1 Sub-Nyquist Radar: Principles and Prototypes · Radar remote sensing advanced tremendously over the past several decades and is now applied to diverse areas such as military surveillance,

1 Sub-Nyquist Radar: Principles andPrototypes

Kumar Vijay Mishra1and Yonina C. Eldar

2

Andrew and Erna Viterbi Faculty of Electrical Engineering, Technion - Israel Institute of Technology,

Haifa, Israel

1.1 Introduction

Radar remote sensing advanced tremendously over the past several decades and

is now applied to diverse areas such as military surveillance, meteorology, geology,

collision-avoidance, and imaging [1]. In monostatic pulse Doppler radar systems,

an antenna transmits a periodic train of known narrowband pulses within a

defined coherent processing interval (CPI). When the radiated wave from the

radar interacts with moving targets, the amplitude, frequency, and polarization

state of the scattered wave change. By monitoring this change, it is possible

to infer the targets’ size, location, and radial Doppler velocity. The reflected

signal received by the radar antenna is a linear combination of echoes from

multiple targets; each of these is an attenuated, time-delayed, and frequency-

modulated version of the transmit signal. The delay in the received signal is

linearly proportional to the target’s range or its distance from the radar. The

frequency modulation encodes the Doppler velocity of the target. The complex

amplitude or target’s reflectivity is a function of the target’s size, geometry,

propagation, and scattering mechanism. Radar signal processing is aimed at

detecting the targets and estimating their parameters from the output of this

linear, time-varying system.

Traditional radar signal processing employs matched filtering (MF) or pulse

compression [2] in the digital domain wherein the sampled received signal is cor-

related with a replica of the transmit signal in the delay-Doppler plane. The MF

maximizes the signal-to-noise ratio (SNR) in the presence of additive white Gaus-

sian noise. In some specialized systems, this stage is replaced by a mismatched fil-

ter with a different optimization metric such as minimization of peak-to-sidelobe-

ratio of the output. Here, the received signal is correlated with a signal that is

close but not identical to the transmit signal [3–5]. While all of these techniques

reliably estimate target parameters, their resolution is inversely proportional to

the support of the ambiguity function of the transmit pulse thereby restricting

ability to super-resolve targets that are closely spaced.

1 K.V.M. acknowledges partial support via Andrew and Erna Finci Viterbi PostdoctoralFellowship and Lady Davis Postdoctoral Fellowship.

2 This work is supported by the European Unions Horizon 2020 research and innovation

program under grant agreement no. 646804-ERC-COG-BNYQ.

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2 Sub-Nyquist Radar: Principles and Prototypes

The digital MF operation requires the signal to be sampled at or above the

Nyquist sampling rate which guarantees perfect reconstruction of a bandlimited

analog signal [6]. Many modern radar systems use wide bandwidths, typically

ranging from hundreds of MHz to even GHz, in order to achieve fine radar range

resolution. Since the Nyquist sampling rate is twice the baseband bandwidth, the

radar receiver requires expensive, high-rate analog-to-digital converters (ADCs).

The sampled signal is then also processed at high rates resulting in significant

power, cost, storage, and computational overhead. Recently, in order to mitigate

this rate bottleneck, new methods have been proposed that sample signals at

sub-Nyquist rates and yet are able to estimate the targets’ parameters [6, 7].

Analogous trade-offs arise in other aspects of radar system design. For exam-

ple, the number of transmit pulses governs the resolution in Doppler velocity.

The estimation accuracy of target parameters is greatly affected by the radar’s

dwell time [1], i.e., the time duration a directional radar beam spends hitting

a particular target. Long dwell times imply a large number of transmit pulses

and, therefore, high Doppler precision. But, simultaneously, this negatively af-

fects the ability of the radar to look at targets in other directions. Sub-Nyquist

sampling approaches have, therefore, been suggested for the pulse dimension or

“slow time” domain in order to break the link between dwell time and Doppler

resolution [8–10].

Finally, radars that deploy antenna arrays deal with similar sampling prob-

lems in the spatial domain. A phased array radar antenna consists of several

radiating elements that form a highly directional radiating beam pattern. With-

out requiring any mechanical motion, a phased array accomplishes beam-steering

electronically by adjusting the relative phase of excitation in the array elements.

The operational advantage is the agile scanning of the target scene, ability to

track a large number of targets, and efficient search-and-track in the regions-of-

interest [11]. The beam pattern of individual array elements, array geometry, and

its size define the overall antenna pattern [12,13] wherein high spatial resolution

is achieved by large array apertures. As per the Nyquist Theorem, the array must

not admit less than two signal samples per spatial period (i.e., radar’s operating

wavelength) [14]. Otherwise, it introduces spatial aliasing or multiple beams in

the antenna pattern, thereby reducing its directivity. Often an exceedingly large

number of radiating elements are required to synthesize a given array aperture in

order to enhance the radar’s ability to unambiguously distinguish closely spaced

targets; the associated cost, weight, and area may be unacceptable. It is therefore

desirable to apply sub-Nyquist techniques to thin a huge array without causing

degradation in spatial resolution [15–17].

Sub-Nyquist sampling leads to the development of low-cost, power-efficient,

and small size radar systems that can scan faster and acquire larger volumes

than traditional systems. Apart from design benefits, other applications of such

systems have been envisioned recently including imparting hardware feasible cog-

nitive abilities to the radar [18, 19], a role in devising spectrally coexistent sys-

1.2 Prior Art and Historical Notes 3

tems [20], and extension to imaging [21]. In this chapter, we provide an overview

of sub-Nyquist radars, their applications and hardware realizations.

The outline of the chapter is as follows. In the next section, we overview vari-

ous reduced-rate techniques for radar system design and explain the benefits of

our approach to sub-Nyquist radars. In Section 1.3, we describe the principles,

algorithms, and hardware realization of temporal sub-Nyquist monostatic pulse-

Doppler radar. Section 1.4 presents an extension of the sub-Nyquist principle to

slow-time. We then introduce the cognitive radar concept based on sub-Nyquist

reception in Section 1.5 and show an application to coexistence in a spectrally

crowded environment. Section 1.6 is devoted to spatial sub-Nyquist applications

in multiple-input-multiple-output (MIMO) array radars. Finally, we consider

sub-Nyquist synthetic aperture radar (SAR) imaging in Section 1.7, followed

by concluding remarks in Section 1.8.

1.2 Prior Art and Historical Notes

There is a large body of literature on reduced rate sampling techniques for radars.

Most of these works employ compressed sensing (CS) methods which allow re-

covery of sparse, undersampled signals from random linear measurements [7]. A

pre-2010 review of selected applications of CS-based radars can be found in [22].

A qualitative, system-level commentary from the point of view of operational

radar engineers is available in [23] while CS-based radar imaging studies are sum-

marized in [24]. An excellent overview on sparsity-based SAR imaging methods

is provided in [25]. The review in [26] recaps major developments in this area

from a non-mathematical perspective. In the following, we review the most sig-

nificant works relevant to the sub-Nyquist formulations presented in this chapter.

On-grid CS The earliest application of CS to recover time-delays with sub-

Nyquist samples in a noiseless case was formulated in [27]. CS-based parameter

estimation for both delay and Doppler shifts was proposed in [28] with samples

acquired at the Nyquist rate. These and similar later works [29–31] discretize the

delay-Doppler domain assuming that targets lie on a grid. Subsequently, these

ideas were extended to colocated [32,33] and distributed [34] MIMO radars where

targets are located on an angle-Doppler-range grid. In practice, target parame-

ters are typically continuous values whose discretization may introduce gridding

errors [35]. In particular, [28] constructs a dictionary that exhaustively considers

all possible delay-Doppler pairs thereby rendering the processing computation-

ally expensive. Noise and clutter mitigation are not considered in this literature.

Simulations show that such systems typically have poor performance in clutter-

contaminated noisy environments.

Off-grid CS A few recent works [36,37] formulate the radar parameter estima-

tion for off-grid targets using atomic norm minimization [38,39]. However, these


methods do not address direct analog sampling, presence of noise and clutter.

Further details on this approach are available in Chapter 7 (“Super-resolution

radar imaging via convex optimization”) of this book.

Parametric Recovery A different approach was suggested in [40] which treated

radar parameter estimation as the identification of an underlying linear, time-

varying system [41]. The proposed two-stage recovery algorithm, largely based

on [42], first estimates target delays and then utilizes these recovered delays to

estimate Doppler velocities, and complex reflectivities. They also provide guaran-

tees for system identification in terms of the minimum time-bandwidth product

of the input signal. However, this method does not handle noise well.

Matrix Completion In some radar applications, the received signal samples

are processed as data matrices which, under certain conditions, are low rank. In

this context, general works have suggested retrieving the missing entries using

matrix completion methods [10, 43]. The target parameters are then recovered

through classic radar signal processing. These techniques have not been exhaus-

tively evaluated for different signal scenarios and their practical implementations

have still not been thoroughly examined.

Finite-Rate-of-Innovation (FRI) Sampling The received radar signal from

L targets can be modeled with 3L degrees of freedom because three parame-

ters - time delay, Doppler shift, and attenuation coefficient - characterize each

target. The classes of signals that have finite degrees of freedom per unit of

time are called finite-rate-of-innovation (FRI) signals [44]. Low-rate sampling

and signal recovery strategies for FRI signals have been studied in detail in the

past [6, Chapter 15]. In [45], a temporal sub-Nyquist radar was proposed to re-

cover delays relying on the FRI model. The Xampling framework [6] was used

to obtain Fourier coefficients from low-rate samples with a practical hardware

prototype. Similar techniques were later studied for delay channel estimation

problems in ultra-wideband communication systems [46, 47] and for ultrasound

imaging [48]. In [49], Doppler focusing was added to the FRI-Xampling frame-

work to recover both delays and Dopplers. Doppler focusing is a narrowband

technique which can be interpreted as low-rate beamforming in the frequency

domain, and was applied earlier to ultrasound imaging [50, 51]. It considers a

chosen center frequency with a band of frequencies around it and coherently

processes multiple echoes in this focused region to estimate the delays from low-

rate samples.

Extensions of Sub-Nyquist Radars The system proposed in [49] reduces

samples only in time and not in the Doppler domain. Since the set of frequencies

for Doppler focusing is usually fixed a priori, the resultant Doppler resolution is

limited by the focusing; it remains inversely proportional to the number of pulses

P as is also the case with conventional radar. In [8], sub-Nyquist processing in

1.3 Temporal Sub-Nyquist Radar 5

Table 1.1 Sub-Nyquist radars and their corresponding reduction domains

Sub-Nyquist System Temporal Doppler Spatial

Monostatic pulsed radar [45] Yes No NoMonostatic pulse Doppler radar [49] Yes No NoReduced time-on-target radar [8] Yes Yes NoMIMO SCS [16] No No YesPhased array SCS [15] No No YesSUMMeR [17] [15] Yes No YesTenDSuR [54] Yes Yes YesSub-Nyquist SAR [21] Yes No No

slow-time was introduced to recover the target range and Doppler by simulta-

neously transmitting few pulses in the CPI and sampling the received signals

at sub-Nyquist rates. Later, [52] proposed a whitening procedure to mitigate

the presence of clutter in a sub-Nyquist radar. Spatial-domain compressed sens-

ing (SCS) was examined for a MIMO array radar in [16] and later for phased

arrays in [15]. Recently, [17] proposed Xampling in time and space to recover

delay, Doppler, and azimuth of the targets by thinning a colocated MIMO ar-

ray and collecting low-rate samples at each receive element. This sub-Nyquist

MIMO radar (SUMMeR) system was also conceptually demonstrated in hard-

ware [18, 53]. The formulation in [54] proposes Tensor-Based 3D Sub-Nyquist

Radar (TenDSuR) that performs thinning in all three domains and recovers the

signal via tensor-based recovery. Finally, an extension to SAR was demonstrated

in [21]. Table 1.1 summarizes these developments.

1.3 Temporal Sub-Nyquist Radar

Consider a standard pulse-Doppler radar that transmits a pulse train

rTX (t) =

P−1∑p=0

h(t− pτ), 0 ≤ t ≤ Pτ, (1.1)

consisting of P uniformly spaced known pulses h(t). The interpulse transmit

delay τ is the pulse repetition interval (PRI) or fast time; its reciprocal is the

pulse repetition frequency (PRF). The entire duration of P pulses in (1.1) is

known as the CPI or slow time. The radar operates at carrier frequency fc so

that its wavelength is λ = c/fc, where c = 3× 108 m/s is the speed of light.

In a conventional pulse Doppler radar, the pulse h(t) = hNyq(t) is a time-

limited baseband function whose continuous-time Fourier transform (CTFT) is

HNyq(f) =∫∞−∞ hNyq(t)e−j2πftdt. It is assumed that most of the signal’s en-

ergy lies within the frequencies ±Bh/2, where Bh denotes the effective signal


bandwidth, such that the following approximation holds:

hNyq(t) ≈Bh/2∫−Bh/2

HNyq(f)ej2πftdf. (1.2)

The total transmit power of the radar is defined as∫ Bh/2

−Bh/2|HNyq(f)|2 df = PT . (1.3)

1.3.1 Received Signal Model

Assume that the radar target scene consists of L non-fluctuating point-targets,

according to the Swerling-0 target model [1]. The transmit signal is reflected

back by the L targets and these echoes are received by the radar processor. The

latter aims at recovering the following information about the L targets from the

received signal: time delay τl, which is linearly proportional to the target’s range

dl = cτl/2; Doppler frequency νl, proportional to the target’s radial velocity

vl = cνl/4πfc; and complex amplitude αl. The target locations are defined with

respect to the polar coordinate system of the radar and their range and Doppler

are assumed to lie in the unambiguous time-frequency region, i.e., the time delays

are no longer than the PRI and Doppler frequencies are up to the PRF.

Typically, the radar assumes the following operating conditions which leads to

a simplified expression for the received signal [49]:

A1 “Far targets” - assuming νl � 2πfcτl/Pτ , target radar distance is large

compared to the distance change during the CPI over which attenuation

αl is allowed to be constant.

A2 “Slow targets” - assuming νl � 2πfc/PτBh, target velocity is small enough

to allow for constant τl during the CPI. In this case, the following

piecewise-constant approximation holds νlt ≈ νlpτ , for t ∈ [pτ, (p+ 1)τ ].

A3 “Small acceleration” - assuming dνl/dt � c/2fc(Pτ)2, target velocity re-

mains approximately constant during the CPI allowing for constant νl.

A4 “No time or Doppler ambiguities” - The delay-Doppler pairs (τl, νl) for all

l ∈ [1, L] lie in the radar’s unambiguous region of delay-Doppler plane

defined by [0, τ ]× [−π/τ, π/τ ].

A5 The pairs in the set (τl, νl) for all l ∈ [1, L] are unique.

Under the above assumptions, the received signal can be written as

rRX (t) =

P−1∑p=0

L−1∑l=0

αlh(t− τl − pτ)e−jνlt + w(t), (1.4)

for 0 ≤ t ≤ Pτ , where w(t) is a zero mean wide-sense stationary random signal

with autocorrelation rw(s) = σ2δ(s). It is convenient to express rRX (t) as a sum


of single frames

rRX (t) =

P−1∑p=0

rpRX (t) + w(t), (1.5)

where

rpRX (t) =

L−1∑l=0

αlh(t− τl − pτ)e−jνlpτ , (1.6)

for pτ ≤ t ≤ (p+ 1)τ , is the return signal from the pth pulse.

A classical radar signal processor samples each incoming frame rpRX (t) at the

Nyquist rate Bh to yield the digitized samples rpRX [n], 0 ≤ n ≤ N − 1, where

N = τBh. The signal enhancement process employs a MF for the sampled frames

rpRX [n]. This is then followed by Doppler processing where a P -point discrete

Fourier transform (DFT) is performed on slow time samples. By stacking all the

N DFT vectors together, a delay-Doppler map is obtained for the target scene.

Finally, the time delays τl and Doppler shifts νl of the targets are located on this

map using, e.g., a constant false-alarm rate detector [55].

1.3.2 Sub-Nyquist Delay-Doppler Recovery

Traditional radar systems sample the received signal at the Nyquist rate, deter-

mined by the baseband bandwidth of h(t). Our goal is to recover rpRX (t) from

its samples taken far below this rate. We note that over the interval τ , rpRX (t)

is completely specified by {αl, τl, νl}Ll=1, and is an FRI signal with rate of in-

novation 3L/τ . Hence, in the absence of noise, one would expect to be able to

accurately recover rpRX (t) from only a few samples per time τ . Since radar signals

tend to be sparse in the time domain, simply acquiring a few data samples at

a low rate will not generally yield adequate recovery. Indeed, if the separation

between samples is larger than the effective spread in time, then with high prob-

ability many of the samples will be close to zero. This implies that presampling

analog processing must be performed on the frequency-domain support of the

radar signal in order to smear the signal in time before low rate sampling.

The approach we adopt follows the Xampling architecture designed for sam-

pling and processing of analog inputs at rates far below Nyquist, whose under-

lying structure can be modeled as a union of subspaces (UoS). The input signal

belongs to a single subspace, a priori unknown, out of multiple, possibly even

infinitely many, candidate subspaces. Xampling consists of two main functions:

low rate analog to digital conversion (ADC), in which the input is compressed

in the analog domain prior to sampling with commercial devices, and low rate

digital signal processing, in which the input subspace is detected prior to digital

signal processing. The resulting sparse recovery is performed using CS techniques

adapted to the analog setting. This concept has been applied to both communi-

cations [56–59] and radar [49,60], among other applications.

Time-varying linear systems, which introduce both time-shifts (delays) and


frequency-shifts (Doppler-shifts), such as those arising in surveillance point-

target radar systems, fit nicely into the UoS model. Here, a sparse target scene

is assumed, allowing to reduce the sampling rate without sacrificing delay and

Doppler resolution. The Xampling-based system is composed of an ADC which

filters the received signal to predetermined frequencies before taking point-wise

samples. These compressed samples, or “Xamples”, contain the information

needed to recover the desired signal parameters.

To demonstrate sub-Nyquist sampling, we begin by deriving an expression

for the Fourier coefficients of the received signal and show that the target pa-

rameters are embodied in them. Let FR and fNyq be the set of all frequencies

in the received signal spectrum and the corresponding Nyquist sampling rate,

respectively. Consider the Fourier series representation of the aligned frames

rpRX (t+ pτ), with rpRX (t) defined in (1.6):

cp[k] =

∫ τ

0

rpRX (t+ pτ)e−j2πkt/τdt =1

τH[k]

L−1∑l=0

αle−j2πkτl/τe−jνlpτ , (1.7)

for k ∈ κ, where κ ={k =

⌊f

fNyqN⌋∣∣∣ f ∈ FR}. From (1.7), we see that the un-

known parameters {αl, τl, νl}L−1l=0 are embodied in the Fourier coefficients cp[k].

We can estimate these parameters using only a small number of Fourier coeffi-

cients which translates to a low sampling rate.

There are several ways to implement a sub-Nyquist sampler [47,61] in order to

obtain a set of Fourier coefficients from low-rate samples of the signal. For sim-

plicity, consider |κ| = K such that q = N/K is an integer defining the sampling

reduction factor. In direct sampling (Fig. 1.1a), the signal rRx(t) obtained after

the anti-aliasing filter is passed through as many analog chains as the number

of sub-Nyquist coefficients K. Each branch is modulated by a complex exponen-

tial, followed by integration over τ and necessary digital signal processing (DSP).

This technique provides the largest flexibility in choosing the Fourier coefficients,

but is expensive in terms of hardware. Another approach is to limit the band-

width of the anti-aliasing filter such that only the lowest K frequencies are free

of aliasing (Fig. 1.1b). We then sample these lowest K frequencies. Here, the

measurements are correlated and a modification in the analog hardware is also

required so that the anti-aliasing filter has reduced passband. In the multiband

sampling method shown in Fig. 1.1c, M disjoint randomly-chosen groups of con-

secutive Fourier coefficients are obtained such that the total number of sampled

coefficients is K. This translates to splitting the signal across M branches, pass-

ing the downconverted signal through reduced-passband anti-aliasing filters, and

then sampling each band with a low-rate ADC. This method can be easily imple-

mented but requires M low-rate ADCs. The sub-Nyquist hardware prototypes

developed in [45,49] adopt multiband sampling using four groups of consecutive

coefficients. In practice, the specific Fourier coefficients are chosen through ex-

tensive software simulations to provide low mutual coherence [6] for CS-based

signal recovery.


Figure 1.1 Sub-Nyquist sampling methods: (a) direct sampling (b) low frequenciesonly (c) multiband sampling.

Figure 1.2 Sum of exponents |g(ν|νl)| for P = 200,τ = 1sec, and νl = 0 [20, 49]. c©2018 IEEE. Reprinted,with permission, from D. Cohen, K. V. Mishra, and Y.C. Eldar, “Spectrum sharing radar: Coexistence viaXampling,” IEEE Transactions on Aerospace andElectronic Systems, vol. 29, no. 3, pp. 1279-1296,2018.

Our goal now is to recover {αl, τl, νl}L−1l=0 from cp[k] for k ∈ κ and 0 ≤ p ≤ P−1.

To that end, [49] adopts the Doppler focusing approach. Consider the DFT of

the coefficients cp[k] in the slow time domain:

Ψν [k] =

P−1∑p=0

cp[k]ejνpτ =1

τH[k]

L−1∑l=0

αle−j2πkτl/τ

P−1∑p=0

ej(ν−νl)pτ . (1.8)

The key to Doppler focusing follows from the approximation:

g(ν|νl) =

P−1∑p=0

ej(ν−νl)pτ ≈{P |ν − νl| < π/Pτ

0 |ν − νl| ≥ π/Pτ,(1.9)

as illustrated in Fig. 1.2. Denote the normalized focused measurements by

Ψν [k] =τ

PH[k]Ψν [k]. (1.10)

As in traditional pulse Doppler radar, suppose we limit ourselves to the Nyquist

grid so that τl/τ = rl/N , where rl is an integer satisfying 0 ≤ rl ≤ N − 1. Then,

(1.10) can be approximately written in vector form as

Ψν = Fκxν , (1.11)

where Ψν = [Ψν [k0] . . .Ψν [kK−1]] , ki ∈ κ for 0 ≤ i ≤ K − 1, Fκ is composed


of the K rows of the N × N Fourier matrix indexed by κ and xν is a L-sparse

vector that contains the values αl at the indices rl for the Doppler frequencies

νl in the “focus zone”, that is |ν − νl| < π/Pτ . It is convenient to write (1.11)

in matrix form, by vertically concatenating the vectors Ψν , for ν on the Nyquist

grid, namely ν = − 12τ + 1

Pτ , into the K × P matrix Ψ, as

Ψ = FκX, (1.12)

where X is formed similarly by vertically concatenating the vectors xν . Note that

the matrix Fκ is not square and, as a result, the system of linear equations (1.12)

is under-determined. The system in (1.12) can be solved using any CS algorithm

such as orthogonal matching pursuit (OMP) and `1 minimization [6, 7].

A Nyquist receiver needs Bhτ samples to recover the targets. However, as

stated in Theorem 1.3.1 below, the number of samples required by the sub-

Nyquist receiver depends only on the number of targets present and not on Bh.

This shows that a sub-Nyquist radar breaks the link between range resolution

and transmit bandwidth. In general, only a few targets are present in the radar

coverage region leading to a significant reduction in sampling rate.

Theorem 1.3.1. [49] The minimal number of samples required for perfect recov-

ery of {αl, τl, νl}Ll=0 in a noiseless environment is 4L2. In addition, the number

of samples per period is at least 2L, and the number of periods P ≥ 2L.

The Doppler focusing operation (1.8) is a continuous operation on the vari-

able ν, and can be performed for any Doppler frequency up to the PRF. With

Doppler focusing there are no inherent blind speeds, i.e., target velocities which

are undetectable, as occurs with classic moving target indicator [1]. Since strong

amplitudes are indicative of true target existence as opposed to noise, Doppler

focusing recovery searches for large magnitude entries and marks them as de-

tections. After detecting each target, its influence is removed from the set of

Fourier coefficients in order to reduce masking of weaker targets and to remove

spurious targets created by processing sidelobes. It is important to note that our

dictionary in (1.12) is indifferent to the Doppler estimation. CS methods which

estimate delay and Doppler simultaneously [28], require a dictionary which grows

with the number of pulses. Here by separating delay and Doppler estimation, the

CS dictionary is not a function of P .

Moreover, the performance of the sub-Nyquist radar in the presence of noise

improves with Doppler focusing. The following theorem states that Doppler fo-

cusing increases the per-target SNR by a factor of P . This linear scaling is similar

to that obtained using a MF.

Theorem 1.3.2. [49] Let the pre-focusing SNR of the lth target be Γlp[k] =|clp[k]|2

E[|wp[k]|2] where clp[k] and wp[k] are the signal and white noise Fourier coeffi-

cients. Then, the focused SNR for the lth target at center frequency ν is PΓlp[k].

A continuous-value parameter recovery using Doppler focusing is described

in [49]. For practical considerations of computational efficiency, Doppler focusing


can be performed on a uniform grid of frequencies so that focused coefficients

are computed efficiently using the fast Fourier Transform (FFT). Algorithm 1

outlines this approach to solving the P equations (1.12) simultaneously where,

in each iteration, the maximal projection of the observation vectors onto the

measurement matrix are retained. The algorithm termination criterion follows

from the generalized likelihood ratio test (GLRT) framework presented in [62].

For each iteration, the alternative and null hypotheses in the GLRT problem

define the presence or absence of a candidate target, respectively. In Algorithm 1,

Qχ22(ρ) denotes the right-tail probability of the chi-square distribution function

with 2 degrees of freedom, ΛC is the complementary set of Λ and

ρ =PT

σ2|FR|(1.13)

is the SNR with σ2 the noise variance and PT the total transmit power.

In Section 1.3.4, we introduce a sub-Nyquist prototype implementing the ideas

in this section using simple hardware. Before that, we describe how to account

for clutter mitigation in sub-Nyquist radar.

1.3.3 Sub-Nyquist Clutter Removal

Clutter refers to unwanted echoes from stationary objects such as buildings,

trees, chaff, and ground surface as well as moving elements like weather and

sea. Since strong clutter echoes hamper detection of desired targets, clutter re-

moval has been investigated intensively. In the context of CS-based radars, [63]

provides a general overview of clutter rejection algorithms. In [64], Capon beam-

forming is used to reject clutter and then the target is retrieved by exploiting

sparse reconstruction methods. On the other hand, a few works such as [65–67]

utilize sparsity of the clutter in the mitigation process. Along similar lines, [68]

assumes sparse clutter and proposes a GLRT detector. However, they obtain

signal samples at the Nyquist rate.

Conventionally, clutter is modeled as a random process with Doppler frequency

that follows a colored Gaussian noise distribution [69–71]. A standard operation

to remove this correlated noise is to use receive filters that maximize the signal-

to-clutter-plus-noise (SCNR) ratio. This method is equivalent to first whitening

the received signal samples, and then performing matched-filtering with respect

to a whitened pulse. Our approach [52] to clutter removal in sub-Nyquist radar

is based on this philosophy as it fits well with our Fourier-based analysis.

In the presence of clutter and noise, the received signal rq(t) is

r(t) = rRX (t) + y(t), (1.14)

where rRX (t) is the target signal with noise as in (1.5) and

y(t) =

P−1∑p=0

C∑c=1

αch(t− pτ − τc)ejvcpτ , (1.15)

is the echo from C clutter targets. We assume that the mean clutter amplitude is


Algorithm 1 Sub-Nyquist Radar Delay-Doppler Recovery [20,49]

Input: Observations cp[k], 0 ≤ p ≤ P − 1 and k ∈ κ, probability of false alarm

Pfa, noise variance σ2, transmitted power PT , total transmitted bandwidth

|FR|Output: Estimated target parameters {αl, τl, νl}L−1

l=0

1: Create Ψ from cp[k] using the FFT (1.8), for k ∈ κ and ν = −1/(2τ) +

p/(Pτ), 0 ≤ p ≤ P − 1

2: Compute detection thresholds

ρ =PT

σ2|FR|, γ = Q−1

χ22(ρ)

(1− N√

1− Pfa)

3: Initialization: residual R0 = Ψ, index set Λ0 = ∅, t = 1

4: Project residual onto measurement matrix:

Φ = FHκ Rt−1

5: Find the two indices λt = [λt(1) λt(2)] such that

[λt(1) λt(2)] = arg maxi,j |Φi,j |

6: Compute the test statistic

Γ =(Fκ)λt(1)((Rt−1)λt(2))

H((Fκ)λt(1))H(Rt−1)λt(2)

σ2

where (M)i denotes the ith column of M

7: If Γ > γ continue; otherwise go to step 12

8: Augment index set Λt = Λt⋃{λt}

9: Find the new signal estimate

Xt|Λt = (Fκ)†ΛtΨ, Xt|ΛCt = 0

10: Compute new residual

Rt = Ψ− (Fκ)ΛtX

11: Increment t and return to step 4

12: Estimated support set Λ = Λt13: τl = τ

N Λ(l, 1), νl = 1Pτ Λ(l, 2), αl = XΛ(l,1),Λ(l,2)

E[|αc|2] = σ2c . Further, the delays τc ∼ U(0, τ) and the clutter Doppler spectrum

vc ∼ N (vd, σ2d) are independent and identically distributed.

Analogous to the target signal in (1.7), the Fourier series representation of the

clutter signal is given by

cp[k] =1

τH[k]

C−1∑c=0

αce−j 2π

τ kτce−jvcpτ . (1.16)

Let the Fourier series coefficients of the noise be wp[k]. We now form a P ×Kmatrix R with kth column given by the Fourier coefficients Rp[k] = cp[k]+cp[k]+


wp[k], k ∈ κ such that

R = X + Y + N = FPAFKNH + B, (1.17)

where B = Y+N, FP is the P ×P Fourier matrix with (l, k)th element e−j2πP lk,

FKN is a submatrix formed by K rows of the N ×N Fourier matrix with (l, k)th

element ej2πN lk, H = diag(H[k]) is a K × K diagonal matrix, A is a P × K

sparse matrix with complex reflectivity αl at the L indices (rl, sl), Y and N are

P×K matrices with (p, k)th elements cp[k] and wp[k], respectively. As mentioned

previously, noise is white over the indices k (all tones).

Our goal is to extract A from the measurements R. For simplicity, we assume

that |H[k]|2 is unity for all k. The whitening transformation requires information

about the statistics of clutter and noise, which are summarized in the following

proposition.

Proposition 1.3.3. [52] The mean of the clutter Fourier coefficients is E[cp[k]

]=

0, and their correlation is given by

Rl1 [k1, k2] = E[cp[k1]cp+l1 [k2]

]= Cσ2

cδk1,k2e−jvdl1τ−

12σ

2dl

21τ

2

. (1.18)

The mean and variance of the Fourier coefficients of the noise are, respectively,

E[Np[k]

]= 0, E

[Np[k1]Np+l1 [k2]

]=

1

τσ2nδk1k2

δl2 . (1.19)

Our clutter mitigation technique is based on whitening all the tones of the

measurements R. It follows from Proposition 1.3.3 that the columns of B are un-

correlated and identically distributed. The covariance matrix M of the columns

of B is a Toeplitz matrix with mth diagonal value

M(m) = Cσ2ce−jvdmτ− 1

2σ2dm

2τ2

+1

τσ2nδm. (1.20)

Therefore, the columns of R can be whitened by multiplying on the left by

M−1/2:

M−1/2R = M−1/2FPAFKNH + M−1/2B, (1.21)

where M−1/2B corresponds to white noise. From here, we proceed with Doppler

focusing on M−1/2R by taking a Hermitian transpose of (1.21) and multiplying

on the right by M−1/2FP :

Ψ = H(FKN )HAHFHP M−1FP + BHM−1FP = Ψ + W (1.22)

where W is white noise for each focused frequency. This equation represents a

sparse matrix recovery problem. For known matrices D1 = H(FKN )H and D2 =

FHP M−1FP , we are given measurements Ψ = D1XD2, and the goal is to retrieve

the sparse matrix X = AH . These problems are solved by matrix sketching

algorithms as described in [72]. It has been shown in [52] that whitened Doppler

focusing generally increases the SCNR [52]. Compared to other CS-based radars

[27, 28], this technique is robust to the presence of clutter despite sampling at

low-rates.


Figure 1.3 The four-channel Xamplerboard [45]. c©2014 IEEE. Reprinted, withpermission, from E. Baransky, G. Itzhak,I. Shmuel, N. Wagner, E. Shoshan, andY. C. Eldar, “A sub-Nyquist radarprototype: Hardware and algorithms,”IEEE Transactions on Aerospace andElectronic Systems, vol. 50, pp. 809-822,2014.

Figure 1.4 Sub-Nyquist hardwareprototype showing connections betweenthe Xampler board and NIchassis [8, 45, 49]. c©2016 IEEE.Reprinted, with permission, from D.Cohen and Y. C. Eldar, “Reducedtime-on-target in pulse Doppler radar:Slow time domain compressed sensing,” inIEEE Radar Conference, 2016, pp. 1-4.

1.3.4 Sub-Nyquist Hardware Prototype

The first sub-Nyquist radar hardware implementation was presented in [45]. It

was then developed further to incorporate Doppler focusing and clutter removal

in [49] and [52], respectively. Since sub-Nyquist techniques manifest themselves

mostly in the radar receiver, this prototype emulates receive processing.

The basic prototype (Fig. 1.4) consists of an analog front-end (Fig. 1.3), fed by

a synthetized RF signal using National Instruments (NI) hardware and followed

by digital delay-Doppler map recovery. To evaluate the Xampler board, we make

use of NI equipment for both system synchronization and RF signal sources.

Figure 1.5 shows the entire assembly wrapped in the NI chassis. We transmit 50

pulses with bandwidth 20 MHz. At the receiver, a multiple bandpass sampling

approach was chosen, where four groups of consecutive Fourier coefficient subsets

are selected. Each channel is fed by a local oscillator (LO), which modulates the

desired frequency band of the channel to the central frequency of a narrow 80

KHz bandwidth band pass filter (BPF). A fifth LO, common to all 4 channels,

modulates the BPF output to a low frequency band, and sampled with a standard

low rate ADC operating at 250 kHz frequency. The digital samples are acquired

by the chassis controller and a MATLAB function is launched that runs Doppler

focusing. The digital reconstruction algorithm, performed at a low rate of 250

kSps, allows recovery of the unknown delays and Doppler frequencies of the

targets. A block diagram of the system is shown in Fig. 1.6.

Real-time analog experiments show that the system is able to maintain good

detection capabilities, while sampling radar signals that require Nyquist rate of


Figure 1.5 NIchassis showingvarious signalgeneration andsynchronizationcomponents.

Figure 1.6 Block diagramof 4-channel solid-statereceiver with fourup-modulating localoscillators with respectivecenter frequencies of 28.375MHz, 28.275 MHz, 27.65MHz, and 27.391 MHz [45].c©2014 IEEE. Reprinted,

with permission, from E.Baransky, G. Itzhak, I.Shmuel, N. Wagner, E.Shoshan, and Y. C. Eldar,“A sub-Nyquist radarprototype: Hardware andalgorithms,” IEEETransactions on Aerospaceand Electronic Systems, vol.50, pp. 809-822, 2014.

about 30 MHz at a total rate of 1 MHz, i.e., 1/30th of the Nyquist rate. We

conducted several experiments in order to test the accuracy of our system under

various conditions. For example, Fig. 1.7 shows results for a hardware experiment

where the target scene has seven scatterers with different delays and Doppler

frequencies. A few cases of closely spaced targets in the delay-Doppler plane are

also included. Clutter is also added to the scene and identified by the system.

Our low-rate processing rejects the clutter and successfully detects only targets

in the delay-Doppler plane despite sampling at 1/30th of the Nyquist rate. The

digital recovery algorithm is efficient as it involves only solving 1D delay recovery

problems post FFT-based Doppler focusing and without increasing the size of

the dictionary.


Figure 1.7 Sub-Nyquist prototype experiment [52]. Top left to right: Signalcorresponding to targets, clutter, noise, and all three combined. Bottom left to right:Low rate samples at receiver and delay-Doppler map with true and recovered targets.

1.4 Doppler Sub-Nyquist Radar

The temporal sub-Nyquist processing in the fast time domain described in the

previous section breaks the link between signal bandwidth, sampling rate, and

range resolution. The Xampling framework can also be extended in the slow

time or Doppler frequency domain. The Doppler resolution in classical radar

processing is given by 2π/P1τ where P1 is the number of pulses transmitted

during the CPI. In Doppler domain sub-Nyquist processing, we non-uniformly

transmit P2 < P1 pulses and reduce the power consumption and dwell time

in a particular direction without loss of Doppler resolution. The advantage is

gaining the ability to look at other directions within the same CPI by interleaving

transmissions in different directions.

A few other CS-based works [9, 10] have considered reduced time-on-target

(RToT) scenarios without addressing analog sampling. The Doppler sub-Nyquist

processing that we review here was introduced in [8], and is based on the proto-

type and principles presented in the previous section.

We consider a non-uniformly transmitting pulse-Doppler radar such that the

pth pulse is sent at time mpτ , where {mp}P2−1p=0 is an ordered set of integers such

that mp ≥ p. Then, (1.1) is written as

rTX (t) =

P2−1∑p=0

h(t−mpτ), 0 ≤ t ≤ P1τ. (1.23)

The received signal rRX (t) is accordingly expressed as a sum of single frames

rRX (t) =

P2−1∑p=0

rpRX (t), (1.24)

1.4 Doppler Sub-Nyquist Radar 17

where

rpRX (t) =

L−1∑l=0

αlh(t− τl −mpτ)e−jνlmpτ , (1.25)

for 0 ≤ t ≤ P1τ , is the return signal from the pth pulse. Our goal is to recover

the targets range and Doppler frequency from the received signals rpRX (t), with

reduced number of transmit pulses P2 < P1 as well as low-rate samples per pulse.

1.4.1 Xampling in CPI and Delay-Doppler Recovery

As before, we consider the Fourier series representation of the aligned frames

rpRX (t+mpτ):

Xp[k] =1

τH[k]

L−1∑l=0

αle−j2πkτl/τe−jνlmpτ , 0 ≤ k ≤ N − 1, (1.26)

where N = Bhτ . From (1.26), the Fourier coefficients embody all the information

about the unknown parameters {αl, τl, νl}L−1l−0 . The goal is then to recover these

parameters from Xp[k], 0 ≤ p ≤ P2 − 1. The low rate sampling technique is as

described earlier in Section 1.3.2 but the processing steps to recover the target

parameters is different to account for sub-Nyquist sampling in Doppler.

Let X be the K ×P matrix with pth column given by the Fourier coefficients

Xp[k], k ∈ κ. Then X can be expressed as

X = HFKNA(FP2

P1)T , (1.27)

where H = 1τ diag(H[k]), FKN is a K×N partial Fourier matrix, FP2

P1is a P2×P1

partial Fourier matrix indexed by the values of mp, 1 ≤ p ≤ P2, and A is an

N × P1 sparse matrix with αl values at the L indices {sl, τl}. We would like to

recover A from the measurements X.

The system of linear equations (1.27) can be solved by CS techniques. How-

ever, this problem is different than the temporal sub-Nyquist formulation of

Section 1.3.2 where only the range sensing matrix FKN is a partial DFT. In

Doppler sub-Nyquist radar, both range and Doppler sensing matrices (i.e., FKNand FP2

P1, respectively) are partial DFTs. Analogous to Theorem 1.3.1, we have

the following result for the Doppler sub-Nyquist radar.

Theorem 1.4.1. [8] The minimal number of samples required for perfect recov-

ery of A for L targets in noiseless settings is 4L2. In addition, the number of

samples per period is at least 2L, and the number of periods P2 ≥ 2L.

Note that the number of periods P2 here is for non-uniform transmission while

the minimum number of periods in Theorem 1.3.1 pertain to uniformly spaced

pulses in the CPI. Theorem 1.4.1 indicates the lower limit of rate reduction in

temporal and Doppler domains.

To solve for the sparse matrix A in (1.27) one can use the matrix version

of OMP or `1 minimization [6]. Alternatively, Doppler focusing is still approxi-

mately applicable. The non-uniform discrete Fourier transform of the coefficients


Figure 1.8 RToTsub-Nyquist radarprototype [8]. Top left:Targets at two differentazimuths. Bottom left:Echoes from bothdirections are acquiredvia non-uniform pulses.Top and bottom right:Delay-Doppler mapsshowing reconstructionfor both directions.c©2016 IEEE.

Reprinted, withpermission, from D.Cohen and Y. C. Eldar,“Reducedtime-on-target in pulseDoppler radar: Slowtime domaincompressed sensing,” inIEEE Radar Conference,2016, pp. 1-4.

Xp[k] is

Ψν [k] =

P2−1∑p=0

Xp[k]ejνmpτ =1

τH[k]

L−1∑l=0

αle−j2πkτl/τ

P2−1∑p=0

ej(ν−νl)mpτ . (1.28)

This can be approximated similar to (1.9) and solved by Algorithm 1 described

earlier. However, this is a poor approximation because the P2 points in the sum

of exponents∑P2−1p=0 ej(ν−νl)mpτ are not equally spaced over the unit circle.

1.4.2 RToT Hardware Prototype

A hardware implementation of the RToT concept is described in [8]. It uses the

sub-Nyquist hardware prototype presented in Section 1.3.4. We evaluated the

prototype for a scenario wherein targets are located at two distinct azimuths.

Here, P1 = 50 pulses were chosen such that a quarter of them were sent in one

direction and the rest in another. The target scenario for both is then simultane-

ously recovered within the same original CPI (Fig. 1.8). In this experiment, the

reduction in temporal domain is the same as in Section 1.3.4, i.e., 1/30 of the

Nyquist rate. In the Doppler domain, pulses in the two directions are reduced

by 75% and 25%, respectively.

1.5 Cognitive Sub-Nyquist Radar and Spectral Coexistence 19

1.5 Cognitive Sub-Nyquist Radar and Spectral Coexistence

In the previous two sections, we focused on processing the received signal. The

receiver design in the sub-Nyquist framework can be exploited to also alter the

behavior of the radar transmitter. In this section, we discuss the opportunistic

control of the transmitter to impart cognition to the radar and leverage it for

spectrum sharing applications. For alternative, non-sub-Nyquist approaches to

cognition in radars, we refer the reader to Chapters 9 (“Spectrum sensing for

cognitive radar via model sparsity exploitation”) and 10 (“Cooperative spectrum

sharing between sparse sensing based radar”) of this book.

The unhindered operation of a radar that shares its spectrum with commu-

nication systems has captured a great deal of attention within the operational

radar community in recent years [73]. The interest in such spectrum sharing

radars is largely due to electromagnetic spectrum being a scarce resource and

almost all services having a need for a greater access to it.

Recent research in spectrum sharing radars has focused on S and C-bands,

where the spectrum has seen increasing cohabitation by Long Term-Evolution

(LTE) cellular/wireless commercial communication systems. Many synergistic

efforts by major agencies are underway for efficient radio spectrum utilization.

A significant recent development is the announcement of the Shared Spectrum

Access for Radar and Communications (SSPARC) program [74] by the Defense

Advanced Research Projects Agency (DARPA). This program is focused on S-

band military radars and views spectrum sharing as a cooperative arrangement

where the radar and communication services actively exchange information. It

defines spectral coexistence as equipping existing radar systems with spectrum

sharing capabilities and spectral co-design as developing new systems that utilize

opportunistic spectrum access [75]. For a review of spectral interference from

different services at IEEE radar bands, see [20].

A variety of system architectures have been proposed for spectrum sharing

radars. Most put emphasis on optimizing the performance of either radar or

communications while ignoring the performance of the other. The radar-centric

architectures [20,76] usually assume fixed interference levels from communication

systems and design the system for high probability of detection (Pd). Similarly,

the communications-centric systems attempt to improve performance metrics like

the error vector magnitude and bit/symbol error rate for interference from radar.

With the introduction of the SSPARC program, joint radar-communication per-

formance is being investigated [77]. In nearly all cases, real-time exchange of

information between radar and communications hardware has not yet been in-

tegrated into the system architectures. In a similar vein, our proposed method,

described later in this section, incorporates handshaking of spectral information

between the two systems.

Conventional receiver processing techniques to remove RF interference in radar

employ notch filters at hostile frequencies. Typically, spectrum sharing is achieved

by notching out the radar waveform’s bandwidth causing a decrease in range res-


Figure 1.9 A conventional radarwith bandwidth Bh transmits inthe band Bh. A cognitive radartransmits only in subbands{Bi}Nbi=1, but with increasedin-band power. The sub-Nyquistreceiver samples and processesonly these subbands [19]. c©2017IEEE. Reprinted, with permission,from K. V. Mishra and Y. C.Eldar, “Performance of time delayestimation in a cognitive radar,” inIEEE International Conference onAcoustics, Speech and SignalProcessing, 2017, pp. 3141-3145.

olution. Our spectrum sharing solution departs from this baseline. The approach

we adopt follows the Xampling architecture on which the sub-Nyquist radar pro-

totype described earlier in Section 1.3 is based. We recall that the sub-Nyquist

receiver samples and processes only small narrow subbands of the received sig-

nal. Hence, we capitalize on the simple observation that if only narrow spectral

bands are sampled and processed, then one can restrict the transmit signal to

these bands. The concept of transmitting only a few subbands that the receiver

processes is one way to formulate a cognitive radar (CRr) [60]. The delay-Doppler

recovery is then performed as presented earlier in Section 1.3. The range resolu-

tion obtained through this multiband signal spectrum fragmentation can be the

same as that of a wideband traditional radar. Furthermore, by concentrating all

the available power in the transmitted narrow bands rather than over a wide

bandwidth, the CRr increases SNR as illustrated in Fig. 1.9.

In the CRr system [60], the support of subbands varies with time to allow for

dynamic and flexible adaptation to the environment. Such a system also enables

the radar to disguise the transmitted signal as an electronic counter measure or

cope with crowded spectrum by using a smaller interference-free portion. The

CRr configuration is key to spectrum sharing since the radar transceiver adapts

its transmission to available bands, achieving coexistence with communication

signals. To detect vacant bands, a communication receiver is needed, that per-

forms spectrum sensing over a large bandwidth. Such systems have recently re-

ceived tremendous interest in communications research, which faces a bottleneck

in terms of spectrum availability. To increase the efficiency of spectrum manag-

ing, dynamic opportunistic exploitation of temporarily vacant spectral bands

by secondary users has been considered, under the name of Cognitive Radio

(CRo) [78]. Here, we use a CRo receiver to detect the occupied communica-

tion bands, so that our radar transmitter can exploit the spectral holes. One of

the main challenges of spectrum sensing in the context of CRo is the sampling


~ ~

𝑓1

~ ~

𝑓Nyq

2

−𝑓Nyq

20 𝑓2 𝑓3−𝑓1-𝑓2-𝑓3

𝐵

Figure 1.10 Multibandmodel with K = 6bands [20]. c©2018 IEEE.Reprinted, with permission,from D. Cohen, K. V.Mishra, and Y. C. Eldar,“Spectrum sharing radar:Coexistence via Xampling,”IEEE Transactions onAerospace and ElectronicSystems, vol. 29, no. 3, pp.1279-1296, 2018.

rate bottleneck due to the wide signal bandwidth. In this context, we use the

Xampling framework to subsample and process the signal [20,56].

Denote the set of all frequencies of the available common spectrum by F . The

communication and radar systems occupy subsets FC and FR of F , respectively,

such that FC ∩FR = ∅. Once the CRo receiver has identified FC , it provides the

radar with spectral occupancy information. Equipped with this spectral map as

well as a known radio environment map (REM) detailing typical interference, the

CRr transmitter chooses narrow frequency subbands that minimize interference

for its transmission. The radar conveys the frequencies FR to the communica-

tion receiver as well, so that it can ignore the radar bands while sensing the

spectrum. The combined CRo-CRr system results in spectral coexistence via the

Xampling (SpeCX) framework, which optimizes the radar’s performance without

interfering with existing communication transmissions. Our hardware prototype

for SpeCX presented in Section 1.5.3 performs real-time recovery of CRo and

CRr signals sharing a common spectrum at SNRs as low as −5 dB.

1.5.1 Cognitive Radio

We first introduce the signal model, processing, and prototype of CRo in the

context of SpeCX. Let xC(t) be a real-valued continuous-time communication

signal, supported on F = [−1/2TNyq,+1/2TNyq] and composed of up to Nsig

transmit waveforms such that

xC(t) =

Nsig∑i=1

si(t), (1.29)

where s(t) has unknown carrier frequency fi, and Xc(f) is the Fourier transform

of xC(t). We denote by fNyq = 1/TNyq the Nyquist rate of xC(t). The waveforms,

respective carrier frequencies and bandwidths are unknown. We only assume that

the single-sided bandwidth Bic for the ith transmission does not exceed an upper

limit B. Such sparse wideband signals belong to the so-called multiband signal

model [56, 79]. Figure 1.10 illustrates the two-sided spectrum of a multiband


𝑥 𝑡RF Signal

𝑦𝑖(𝑡)𝑖 = 1. . 3

PC – Matlab Based Controller Generates RF Input in Real Time

*Note: Allows real-time sampling

𝑆𝑖𝑔𝑛𝑎𝑙 𝐴𝐷𝐶

NI PXIe-1065 with DC Coupled 4-Channel ADC

PC - Labview + Matlab Based Controller

The Cog-Radio Card

Mixing Series Generator

XILINX VC707 – HighSpeed FPGA

𝑝𝑖 𝑡

Signal Generators

2x NI© USRP-2942R RF Generator

PCIe x4 to MXI x4

Figure 1.11 CRo system [80].Reprinted/adapted bypermission from SpringerNature: “Analog to digitalcognitive radio,” in Handbook ofCognitive Radio, W. Zhang, Ed.by D. Cohen, S. Tsiper, and Y.C. Eldar c©(2017).

signal with K = 2Nsig bands centered around unknown carrier frequencies |fi| ≤fNyq/2.

Let FC ⊂ F be the unknown support of xC(t). The goal of the CRo commu-

nication receiver is to retrieve FC , while sampling and processing xC(t) at low

rates in order to reduce system cost and resources. A CRo system was developed

earlier [56, 80] for blind sensing (see Fig. 1.11). Next, we explain the details on

combining this system with the sub-Nyquist radar to implement SpeCX.

The input signal at the communication receiver of the SpeCX system is

x(t) = xC(t) + xR(t), (1.30)

where xR(t) = rTX (t) + rRX (t) is the radar signal sensed by the communica-

tion receiver, composed of the transmitted and received radar signals defined in

(1.1) and (1.4), respectively. Since the frequency support of xC(t) is unknown,

a classic processor would sample such a signal at its Nyquist rate, which can be

prohibitively high. In this work, we instead use the modulated wideband con-

verter (MWC) [56], a sub-Nyquist sampling technique that achieves the lower

sampling rate bound for perfect blind recovery of multiband signals, namely

twice the Landau rate, and is also practically feasible. The MWC is composed of

M parallel channels. In each channel, an analog mixing front-end, where xC(t)

is multiplied by a mixing function pi(t), aliases the spectrum, such that each

band appears in baseband. The mixing functions pi(t) are periodic with period

Tp such that fp = 1/Tp ≥ B and have thus the following Fourier expansion:

pi(t) =

∞∑l=−∞

cilej 2πTplt. (1.31)

In each channel, the signal next goes through a lowpass filter (LPF) with cut-off

frequency fs/2 and is sampled at rate fs ≥ fp, resulting in samples zi[n]. Define

N = 2⌈fNyq+fs

2fp

⌉and Fs = [−fs/2, fs/2]. Following the calculations in [56], the

relation between the known discrete time Fourier transform of the samples zi[n]

and the unknown XC(f) is given by

z(f) = A(xC(f) + xR(f)), f ∈ Fs, (1.32)

where z(f) is a vector of length M with ith element zi(f) = Zi(ej2πfTs) and the

unknown vector xC(f) is given by

xCi(f) = XC(f + (i− dN/2e)fp), f ∈ Fs, (1.33)


for 1 ≤ i ≤ N . The vector xRi(f) is defined similarly. The M × N matrix A

contains the known coefficients cil such that Ail = ci,−l = c∗il.

The MWC analog mixing front-end, shown in Fig. 1.12, results in folding the

spectrum to baseband with different weights for each frequency interval. The

CRo’s goal is now to recover the support of xC(f) from the low rate samples

z(f). The recovery of xC(f) for each f independently is inefficient and not robust

to noise. Instead, the support recovery paradigm from [56] exploits the fact that

the bands occupy continuous spectral intervals so that xC(f) are jointly sparse

for f ∈ Fp. The continuous to finite block [56] then produces a finite system of

equations, called multiple measurement vectors (MMV) from the infinite number

of linear systems (1.32).

From (1.32), we have

Q = ΦZΦH , (1.34)

where

Q =

∫f∈Fp

z(f)zH(f)df, Z =

∫f∈Fp

x(f)xH(f)df, (1.35)

are M ×M and N × N matrices, respectively. Here, x(f) = xC(f) + xR(f).

The matrix Q is then decomposed to a frame V such that Q = VVH . Clearly,

there are many ways to select V. One possibility is to construct it by perform-

ing an eigendecomposition of Q and choosing V as the matrix of eigenvectors

corresponding to the non zero eigenvalues. The finite dimensional MMV system

is then given by

V = A(UC + UR). (1.36)

The support of the unique sparsest solution of (1.36) is the same as the support

of our original set of equations (1.32) [56]. Therefore, the support of UC and UR

are disjoint.

The frequency support FR of xR(t) is known at the communication receiver.

From FR, we derive the support SR of the radar slices xR(f), which is identical

to the support of UR, such that

SR =

{n

∣∣∣∣∣∣∣∣n− f iRfp− dN/2e

∣∣∣∣ < fs +BiR2fp

}, (1.37)

for 1 ≤ i ≤ Nb. Our goal can then be stated as recovering the support of

UC from V, given the known support SR of UR. This can be formulated as a

sparse recovery with partial support knowledge, studied under the framework of

modified CS [81]. Modified-CS has been used to adapt CS recovery algorithms to

exploit partial known support. In particular, greedy algorithms, such as OMP,

have been modified to OMP with partial known support [82]. Instead of starting

with an initial empty support set, one starts with SR as the initial support. In

the first iteration, we compute the estimate

USR1 = A†SRV, U1i = 0, ∀i /∈ SR, (1.38)


Figure 1.12 Schematic implementation of the MWC analog sampling front-end anddigital signal recovery from low rate samples [20]. The CRo inputs are thecommunication signal xC(t) and radar support FR. The communication supportoutput FC is shared with the radar transmitter. c©2018 IEEE. Reprinted, withpermission, from D. Cohen, K. V. Mishra, and Y. C. Eldar, “Spectrum sharing radar:Coexistence via Xampling,” IEEE Transactions on Aerospace and Electronic Systems,vol. 29, no. 3, pp. 1279-1296, 2018.

and residual

V1 = V −ASRU1. (1.39)

The remainder of the algorithm is then identical to OMP.

Once the overall support SC⋃SR is known, we have

xSC⋃SR [n] = A†SC

⋃SR

z[n], (1.40)

xi[n] = 0, ∀i /∈ SC⋃SR.

Here, xSC⋃SR(f) denotes the vector x(f) reduced to its support, ASC

⋃SR is

composed of the columns of A indexed by SC⋃SR and † is the Moore-Penrose

pseudo-inverse. The occupied communication support is then

FC = {f ||f − (i+ dN/2e)fp| ≤fp2, for all i ∈ SC}. (1.41)

1.5.2 Cognitive Radar

After CRo detects the communication signal support, the CRr transmits a pulse

h(t) in the unused parts of the spectrum. The transmit signal is supported over

Nb disjoint frequency bands, with bandwidths {Bir}Nbi=1 centered around the re-

spective frequencies {f ir}Nbi=1, such that

∑Nbi=1B

ir < Bh. The number of bands

Nb is known to the receiver and does not change during operation. The loca-

tion and extent of the bands Bir and f ir are determined by the radar transmitter

through an optimization procedure to identify the least contaminated bands (see

Section 1.5.2). The resulting transmitted radar signal CTFT is

HR(f) =

{βiHNyq(f), f ∈ F iR, for 1 ≤ i ≤ Nb0, otherwise,

(1.42)

where F iR = [f ir − Bir/2, fir + Bir/2] is the set of frequencies in the ith band

such that FR =⋃Nbi=1 F iR. The parameters βi > 1 are chosen such that the total


transmit power PT of the spectrum sharing radar remains the same as that of

the conventional radar:∫ Bh/2

−Bh/2|HNyq(f)|2 df =

Nb∑i=1

∫Fir

|HR(f)|2 df = PT . (1.43)

The radar identifies an appropriate transmit frequency set FR ⊂ F \FC such

that the radar’s probability of detection Pd is maximized. For a fixed probability

of false alarm Pfa the Pd increases with higher signal to interference and noise

ratio [55]. At the spectrum sharing radar receiver, we employ the sub-Nyquist

approach described in Section 1.3.2, where the delay-Doppler map is recovered

from the subset of Fourier coefficients defined by FR.

Optimal Radar Transmit Bands

We now explain the procedure through which a CRr selects transmit subbands

that have minimal spectral interference. The REM is assumed to be known to

the radar transmitter in the form of typical interfering energy levels with respect

to frequency bands, represented by a vector y ∈ Rq, where q is the number of

frequency bands with bandwidth by , |F|/q. In addition, the information from

the CRo indicates that the radar waveform must avoid all frequencies in the set

FC . Therefore, we set y to be equal to ∞ in these bands. Our goal is to select

subbands from the set F \FC with minimal interference. We do that by seeking

a block-sparse frequency vector w ∈ Rp with unknown block lengths, where p is

the number of discretized frequencies, whose support indicates frequency bands

with low interference for the radar. Each entry of w represents a subband of

bandwidth bw , |F|/p.To this end, we use the structured sparsity framework of [83] based on the

one-dimensional graph sparsity structure whose nodes denote the p frequency

points of w. In order to find the desired block-sparse w, the formulation in [83]

replaces the traditional sparse recovery `0 constraint by a more general term

c(w), referred to as the coding complexity such that c(F ) = g log p+ |F |, where

F ⊂ {1, . . . , p} is a sparse subset of the index set of the coefficients of w and g is

the number of connected regions or blocks of F . This coding complexity, which

accounts for both the number of discretized frequencies |F | and the number of

connected regions g, favors blocks within the graph. In our setting, this reduces to

solving the following optimization problem for finding the block-sparse frequency

vector w with (yinv)i = 1/yi:

minimizew ||yinv −Dw||22 + λc(w), (1.44)

where λ is a regularization parameter and c(w) is defined by c(w) = minF {c(F )|supp(w) ⊂ F}. The matrix D is q × p matrix and maps each discrete frequency

in w to the corresponding band in yinv. That is, the (i, j)th entry of D is equal

to 1 if the jth frequency in w belongs to the ith band in y; otherwise, it is equal

to 0. Problem (1.44) can be solved using structured OMP [83].


Algorithm 2 Cognitive Radar Band Selection [20]

Input: REM vector y and subbands bandwidth by = |F|/q, shared support

F , communication support FC , mapping matrix D, number of discretized

frequencies p, number of bands NbOutput: Block sparse vector w, radar support FR1: Set yi =∞, for each ith subband not in FC and compute (yinv)i = 1/yi2: Initialization F0 = ∅, w = 0, t = 1

3: Find the index λt so that λt = arg maxφ(i), where

φ(i) =||Pi(Dwt−1 − yinv)||22c(i⋃Ft−1)− c(Ft−1)

with Pi = Di(DTi Di)

†DTi

4: Augment index set Ft = λt⋃Ft−1

5: Find the new estimate wt|Ft = D†Ftyinv, wt|FCt = 0

6: If the number of blocks, or connected regions, g(w) > Nb, go to step 7.

Otherwise, return to step 3

7: Remove the last index λt so that Ft = Ft−1 and wt = wt−1

8: Compute the radar support FR =⋃j∈Ft [jbw−|F|/2, (j+1)bw−|F|/2] with

bw = |F|/p

Delay-Doppler RecoveryIn order to recover the delay-Doppler map from only Nb transmitted narrow

bands, CRr employs a sub-Nyquist receiver that we explained earlier in Sec-

tion 1.3.2. The radar receiver first filters the CRr subbands supported on FRand computes the Fourier coefficients of the received signal. Our resulting spec-

trum sharing SpeCX framework is summarized in Algorithm 3.

Algorithm 3 Spectral Coexistence via Xampling (SpeCX) [20]

Input: Communication signal xC(t)

Output: Estimated target parameters {αl, τl, νl}L−1l=0

1: Initialization: perform spectrum sensing at the CRo receiver on xC(t) fol-

lowing the procedure in Section 1.5.1

2: Choose the least noisy subbands for the radar transmit spectrum with respect

to detected FC using Algorithm 2

3: Send FR to communication and radar receivers

4: Perform target delay and Doppler estimation using Algorithm 1

5: Perform spectrum sensing at the communication receiver on x(t) = xC(t) +

xR(t) following the procedure in Section 1.5.1

6: If FC changes, then the radar transmitter goes back to step 2

For time delay estimation, [19] compares the performance of conventional and

cognitive radars using the extended Ziv-Zakai lower bound (EZB). In a conven-


tional radar, the EZB for a single target delay estimate τ0 is

EZBR(τ0) = σ2τ0 · 2Q

(√SNR

2

)+

Γ3/2

(SNR

4

)SNR · F 2 , (1.45)

where Q(·) denotes the right tail Gaussian probability function, Γa(b) is the

incomplete gamma function with parameter a and upper limit b, and F is the

root-mean-square (rms) bandwidth of the full-band signal. The bound for CRr

is given in the following theorem.

Theorem 1.5.1. [19] The extended Ziv-Zakai lower bound (EZB) for delay

estimation in a cognitive radar is

EZBCRr(τ0) = σ2τ0 · 2Q

√SNR

2

+

Γ3/2

(SNR

4

)Nb∑i=1

SNRi · F 2i

, (1.46)

where SNRi and Fi are the in-band SNR and rms bandwidth of the ith subband

and SNR is the total SNR.

As noted in [19], sinceNb∑i=1

Bir ⊂ Bh, we have SNR > SNR for given PT .

Therefore, the SNR threshold for asymptotic performance of EZBCRr is lower

than EZBR. As the noise increases and power remains constant for both radars,

the asymptotic performance of EZBCRr is more tolerant to noise than EZBR.

The multiband design strategy, besides allowing a dynamic form of the trans-

mitted signal spectrum over only a small portion of the whole bandwidth to

enable spectrum sharing, has two additional advantages. First, as we show in

hardware experiments (Section VI.B), our CS reconstruction achieves the same

resolution as traditional Nyquist processing over a significantly smaller band-

width. Second, the entire transmit power is concentrated in small narrow bands.

Therefore, the SNR in the sampled bands is improved which leads to better

parameter estimation as indicated by Theorem 1.5.1.

1.5.3 SpeCX Prototype

Figure 1.13 shows our SpeCX prototype, composed of a CRo receiver and a CRr

transceiver. The CRo hardware realizes the system shown in Fig. 1.12. At the

heart of the system lies our proprietary MWC board [84] that implements the

sub-Nyquist analog front-end receiver. The card first splits the wideband signal

into M = 4 hardware channels, with an expansion factor of q = 5, yielding

Mq = 20 virtual channels after digital expansion. In each channel, the signal is

then mixed with a periodic sequence pi(t), generated on a dedicated FPGA, with

fp = 20 MHz. The sequences are chosen as truncated versions of Gold Codes.

These were heuristically found to give good detection results [85], primarily due

to small bounded cross-correlations within a set.


Radar Display

Comm Display

Signal Generator

Comm Analog Rx

Radar Analog Rx

Comm Digital Rx

Radar Digital Rx

Figure 1.13 Sharedspectrumprototype [20]. Thesystem is composed ofa signal generator, aCRo receiver based onthe MWC, acommunication digitalreceiver, and a CRranalog and digitalreceiver. c©2018 IEEE.Reprinted, withpermission, from D.Cohen, K. V. Mishra,and Y. C. Eldar,“Spectrum sharingradar: Coexistence viaXampling,” IEEETransactions onAerospace andElectronic Systems, vol.29, no. 3, pp.1279-1296, 2018.

Next, the modulated signal passes through a Chebyshev LPF of 7th order

with a cut-off frequency (−3 dB) of 50 MHz. Finally, the low rate analog signal is

sampled by a National Instruments ADC operating at fs = (q+1)fp = 120 MHz,

leading to a total sampling rate of 480 MHz. The digital receiver is implemented

on a National Instruments PXIe-1065 computer with DC coupled ADC. Since the

digital processing is performed at the low rate 120 MHz, very low computational

load is required in order to achieve real time recovery. MATLAB and LabVIEW

platforms are used for digital recovery operations.

The prototype is fed with RF signals composed of up to Nsig = 5 real com-

munication transmissions, namely K = 10 spectral bands with total bandwidth

occupancy of up to 200 MHz and varying support, with Nyquist rate of 6 GHz. To

test the system’s support recovery capabilities, an RF input is generated using

vector signal generators, each producing a modulated data channel with individ-

ual bandwidth of up to 20 MHz, and carrier frequencies ranging from 250 MHz

up to 3.1 GHz. The input transmissions then go through an RF combiner, result-

ing in a dynamic multiband input signal, that enables fast carrier switching for

each of the bands. This input is specially designed to allow testing the system’s

ability to rapidly sense the input spectrum and adapt to changes, as required by

modern CRo and shared spectrum standards, e.g. in the SSPARC program. The

system’s effective sampling rate, equal to 480 MHz, is only 8% of the Nyquist rate

and 2.4 times the Landau rate. The main advantage of the Xampling framework,


a

b

Figure 1.14 SpeCXcommunication systemdisplay [20] showing (a) low ratesamples acquired from oneMWC channel at rate 120 MHz,and (b) digital reconstruction ofthe entire spectrum fromsub-Nyquist samples. c©2018IEEE. Reprinted, withpermission, from D. Cohen, K.V. Mishra, and Y. C. Eldar,“Spectrum sharing radar:Coexistence via Xampling,”IEEE Transactions on Aerospaceand Electronic Systems, vol. 29,no. 3, pp. 1279-1296, 2018.

demonstrated here, is that sensing is performed in real-time from sub-Nyquist

samples for the entire spectral range.

Support recovery is digitally performed on the low rate samples. The prototype

successfully recovers the support of the CRo transmitted bands, as demonstrated

in Fig. 1.14. The signal is then reconstructed in real-time. Reconstruction does

not require interpolation to the Nyquist rate and the active transmissions are

recovered at the low rate of 20 MHz, corresponding to the bandwidth of the slices

z(f) defined in (1.32). By combining spectrum sensing and signal reconstruction,

the MWC serves as two separate communication devices. The first is a state-of-

the-art CRo that performs real time spectrum sensing at sub-Nyquist rates, and

the second is a receiver able to decode multiple data transmissions simultane-

ously, regardless of their carrier frequencies, while adapting to real time spectral

changes.

The CRr system is based on the sub-Nyquist radar receiver board described

in Section 1.3.4. The prototype simulates transmission of P = 50 pulses towards

L = 9 targets. The CRr transmits over Nb = 4 bands, selected according to

the procedure presented in Section 1.5.2, after the spectrum sensing process has

been completed by the communication receiver. We compare the target detection

performance of our CRr with a traditional wideband radar with bandwidth Bh =

20 MHz. The CRr transmitted bandwidth is thus equal to 3.2% of the wideband.

Figure 1.15 shows windows from the graphical user interface (GUI) of our

CRr system. Figure 1.15(a) illustrates the coexistence between the radar trans-

mitted bands (thick curve) and the existing communication bands (thin curve).

The gain in power is demonstrated in Fig. 1.15(b) which plots the wideband

radar spectrum, CRr, and noise. The true and recovered range-Doppler maps

for the CRr (whose transmit signal consists of four disjoint subbands) are shown

in Fig. 1.15(c). All 9 targets are perfectly recovered and clutter is discarded.

Fig. 1.15(d) shows the performance when the four subbands are joined together


Figure 1.15 SpeCX radar display [20]showing (a) coexisting CRo and CRr (b)CRr spectrum compared with thefull-band radar spectrum. Therange-Doppler display of detected andtrue locations of the targets for the caseof (a) CRr (four disjoint bands) and (d)all four transmit subbands togetherforming a contiguous 320 kHz band.c©2018 IEEE. Reprinted, with permission,

from D. Cohen, K. V. Mishra, and Y. C.Eldar, “Spectrum sharing radar:Coexistence via Xampling,” IEEETransactions on Aerospace and ElectronicSystems, vol. 29, no. 3, pp. 1279-1296,2018.

to result in a 320 kHz contiguous band for the radar transmitter. There are

many missed detections and false alarms in this case. Let the true and estimated

ranges of the ith target be di and di, respectively. Then the root-mean-square

localization error (RMSLE) of L targets is given by

RMSLE =

√√√√ 1

L

L∑i=1

(di − di)2. (1.47)

In Fig. 1.15(c)-(d), the RMSLE is shown as follows: CRr (0.34km), 320 kHz

band or 4 adjacent bands with same bandwidth (8.1km), and wideband (1.2km).

The poor resolution of the 4 adjacent bands scenario is due to its small aper-

ture. The native range resolution in case of 2 MHz wideband scenario is 75 m.

In Fig. 1.15(c), the CRr is able to detect 9 targets at locations 6.097, 31.764,

35.046, 35.451, 35.479, 81.049, 81.570, 121.442, and 120.922 km. Here, the dis-

tance between two closely spaced targets is less than 75 m.

1.6 Spatial Sub-Nyquist: Application to MIMO Radar

We now consider extending sub-Nyquist processing to the spatial domain for

the particular case of MIMO radar [86]. MIMO radars use an array of several

transmit and receive antenna elements, with each transmitter radiating a differ-

ent, mutually orthogonal waveform. Waveform orthogonality can be in time, fre-

quency or code. Our system is based on the collocated MIMO configuration [87],

in which the elements are close to each other so that the radar cross-section of a

target appears identical in all elements. The MIMO receiver separates and coher-

1.6 Spatial Sub-Nyquist: Application to MIMO Radar 31

Figure 1.16 Location of transmit(diamonds) and receive (triangles)antenna elements within the samephysical aperture for (a)conventional MIMO array withT = 5 transmitters and R = 4receivers, (b) virtual ULA withTR = 20 antenna elements, and (c)randomly thinned MIMO array withM = 4 transmitters and Q = 3receivers.

ently processes the target echoes corresponding to each transmitter. The angular

resolution of MIMO using the classic virtual ULA is the same as a phased array

with equivalent virtual aperture but many more antenna elements.

Conventional MIMO radar’s spatial (angular) and range resolutions are lim-

ited by the number of elements and the receiver sampling rate, respectively.

Here, we extend the Xampling framework for temporal sub-Nyquist radar in

Section 1.3 to both space and time by simultaneously thinning an antenna array

and sampling received signals at sub-Nyquist rates. This sub-Nyquist collocated

MIMO radar (SUMMeR) recovers the target range, azimuth, and Doppler ve-

locity without loss of any of the aforementioned radar resolutions. In SUMMeR,

the radar antenna elements are randomly placed within the aperture, and signal

orthogonality is achieved by frequency division multiplexing (FDM). The FDM-

based sub-Nyquist MIMO mitigates the range-azimuth coupling by randomizing

the element locations in the aperture [88].

1.6.1 Sub-Nyquist Collocated MIMO Radar Model

Let the operating wavelength of the radar be λ and the total number of transmit

and receive elements be T and R respectively. The classic approach to collocated

MIMO adopts a virtual ULA structure, where the receive antennas spaced by λ2

and transmit antennas spaced by Rλ2 form two ULAs (or vice versa). Here, the

coherent processing of a total of TR channels in the receiver creates a virtual

equivalent of a phased array with TR λ2 -spaced receivers and normalized aperture

Z = TR2 . This standard array structure and the corresponding receiver virtual

array are illustrated in Fig. 1.16(a)-(b) for T = 5 and R = 4.

Consider a collocated MIMO radar system that has M < T transmit and

Q < R receive antennas. The locations of these antennas are chosen uniformly

at random within the aperture of the virtual array mentioned above, as in

Fig. 1.16(c). The mth transmitting antenna sends P pulses

sm (t) =P−1∑p=0

hm (t− pτ)ej2πfct, 0 ≤ t ≤ Pτ, (1.48)

where {hm (t)}M−1m=0 is a set of narrowband, orthogonal FDM pulses each with


CTFT

Hm(ω) =

∞∫−∞

hm(t)e−jωtdt. (1.49)

For simplicity, we assume that fcτ is an integer. The pulse time support is

denoted by Tp.

Consider a target scene with L non-fluctuating point targets following the

Swerling-0 model [1] whose locations are given by their ranges Rl, Doppler ve-

locity vl, and azimuth angles θl, 1 ≤ l ≤ L. The pulses transmitted by the radar

are reflected back by the targets and collected at the receive antennas. When

the received waveform is downconverted from RF to baseband, we obtain the

following signal at the qth antenna,

xq (t) =

P−1∑p=0

M−1∑m=0

L∑l=1

αlhm (t− pτ − τl) ej2πβmqϑlej2πfDl pτ , (1.50)

where αl denotes the complex-valued reflectivity of the lth target, τl = 2Rl/c is

the range-time delay the lth target, fDl = 2vlc fc is the frequency in the Doppler

spectrum, ϑl = sin θl is the azimuth parameter, and βmq is governed by the array

structure. We express xq(t) as a sum of single frames

xq(t) =

P−1∑p=0

xpq(t), (1.51)

where

xpq(t) =

M−1∑m=0

L∑l=1

αlh(t− τl − pτ)ej2πβmqϑlej2πfDl pτ . (1.52)

Our goal is to estimate the time delay τl, azimuth θl, and Doppler shifts fDlof each target from low rate samples of xq(t), for 0 ≤ q ≤ Q − 1, and a small

number of M channels and Q antennas.

1.6.2 Xampling in Time and Space

The application of Xampling in both space and time enables recovery of range,

direction, and velocity at sub-Nyquist rates. The sampling technique is the same

as in Section 1.3.2 but now the low-rate samples are obtained in both range and

azimuth domains. The received signal xq(t) is separated intoM channels, aligned,

and normalized. The Fourier coefficients of the received signal corresponding to

the channel that processes the mth transmitter echo at the qth receiver are given

by

ypm,q[k] =

L∑l=1

αlej2πβmqϑle−j

2πτ kτle−j2πfmτlej2πf

Dl pτ , (1.53)

where −N2 ≤ k ≤ −N2 − 1, fm is the (baseband) carrier frequency of the mth

transmitter and N is the number of Fourier coefficients per channel.


As in traditional MIMO, assume that the time delays, azimuths, and Doppler

frequencies are aligned to a grid. In particular, τl = τTN sl, ϑl = −1 + 2

TRrl, and

fDl = − 12τ + 1

Pτ ul, where sl, rl ,and ul are integers satisfying 0 ≤ sl ≤ TN − 1,

0 ≤ rl ≤ TR − 1, and 0 ≤ ul ≤ P − 1, respectively. Let Zm be the KQ × Pmatrix with qth column given by the vertical concatenation of ypm,q[k], k ∈ κ, for

0 ≤ q ≤ Q− 1. We can then write Zm as

Zm = (Bm ⊗Am) XDFHP . (1.54)

Here, Am denotes theK×TN matrix whose (k, n)th element is e−j2πTN κkne

−j2π fmBhnT

with κk the kth element in κ, Bm is the Q × TR matrix with (q, p)th ele-

ment e−j2πβmq(−1+ 2TRp), and FP denotes the P ×P Fourier matrix. The matrix

XD is a T 2NR × P sparse matrix that contains the values αl at the L indices

(rlTN + sl, ul).

The range and azimuth dictionaries Am and Bm are not square matrices

due to low-rate sampling of Fourier coefficients at each receiver and reduction

in antenna elements, respectively. Therefore, the system of equations in (1.54)

is undetermined in azimuth and range. Our goal is to recover XD from the

measurement matrices Zm, 0 ≤ m ≤M−1. The temporal, spatial, and frequency

resolution stipulated by XD are 1TBh

, 2TR , and 1

Pτ respectively.

Theorem 1.6.1. [17] The minimal number of transmit and receive array el-

ements, i.e., M and Q, respectively, required for perfect recovery of XD with

L targets in a noiseless setting are determined by MQ ≥ 2L. In addition, the

number of samples per receiver is at least MK ≥ 2L where K is the number of

Fourier coefficients sampled per receiver and the number of pulses per transmitter

is P ≥ 2L.

Theorem 1.6.1 shows that the number of SUMMeR transmit and receive ele-

ments as well as samples K depend only on the number of targets present. These

design parameters, therefore, can be substantially lesser than the requirements

of a Nyquist MIMO array. Similar results for temporal and Doppler sub-Nyquist

radars were obtained in Theorems 1.3.1 and 1.6.1.

1.6.3 Range-Azimuth-Doppler Recovery

To jointly recover the range, azimuth, and Doppler frequency of the targets, we

apply the concept of Doppler focusing from Section 1.3.2 to our MIMO setting.

Doppler focusing for a specific frequency ν yields

Φνm,q[k] =

P−1∑p=0

ypm,q[k]e−j2πνpτ (1.55)

=

L∑l=1

αlej2πβmqϑle−j

2πτ (k+fmτ)τl

P−1∑p=0

ej2π(fDl −ν)pτ ,


for −N2 ≤ k ≤ −N2 − 1. Following Section 1.3.2, it holds thatP−1∑p=0

ej2π(fDl −ν)pτ ∼={P |fDl − ν| < 1

2Pτ ,

0 otherwise.(1.56)

Then, for each focused frequency ν, (1.55) reduces to a 2D problem, which

can be solved using CS recovery techniques, as summarized in Algorithm 4.

Note that step 1 can be performed using the FFT. In the algorithm descrip-

tion, vec(Z) concatenates the columns of Zm, for 0 ≤ m ≤ M − 1, et(l) =[(e0t (l))

T · · · (eM−1t (l))T

]Twhere

emt (l) = vec

((Bm ⊗Am)Λt(l,2)TN+Λt(l,1)

((Fm)TΛt(l,3)

)T), (1.57)

with Λt(l, i) the (l, i)th element in the index set Λt at the tth iteration, and

Et = [et(1) . . . et(t)]. Once XD is recovered, the delays, azimuths, and Dopplers

are estimated as

τl =τΛL(l, 1)

TN, ϑl = −1 +

2ΛL(l, 2)

TR, fDl = − 1

2τ+

ΛL(l, 3)

Pτ. (1.58)

Since in real scenarios, targets delays, Dopplers, and azimuths are not neces-

sarily aligned to a grid, a finer grid can be used around detection points on the

coarse grid to reduce quantization error. This technique adds a step after support

detection in each iteration (step 4 in Algorithm 4).

1.6.4 Multi-Carrier and Cognitive Transmission

The frequency bands left vacant can be exploited to increase the system’s per-

formance without expanding the total bandwidth of Btot = TBh. Denote by

γ = T/M the compression ratio of the number of transmitters. In multi-carrier

SUMMeR, every transmit antenna sends γ pulses, each belonging to a different

frequency band, in one PRI. The total number of user bands is MγBh = TBh.

The ith pulse of the pth PRI is transmitted at time i τγ + pτ , for 0 ≤ i < γ

and 0 ≤ p ≤ P − 1. The samples are then acquired and processed as described

above. Besides increasing the detection performance, this method multiplies the

Doppler dynamic range by a factor of γ with the same Doppler resolution since

the CPI, equal to Pτ , is unchanged. Preserving the CPI allows to maintain the

targets’ stationarity.

Cognitive transmission described in Section 1.5.2 can also be extended to a

SUMMeR system wherein the spectrum of each of the transmitted waveforms

is limited to a few non-overlapping frequency bands while keeping the transmit

power per transmitter the same. Cognitive transmission imparts two advantages

to the SUMMeR hardware. First, the spatial sub-Nyquist processing of large

arrays can be easily designed without replicating the pre-filtering operation for

each subband in the hardware. Second, since the total transmit power remains

the same, a cognitive signal has more in-band power resulting in an increase in

SNR as discussed in Section 1.5.2.


Algorithm 4 Simultaneous sparse 3D recovery based OMP with focusing [17]

Input: Observation matrices Zm, measurement matrices Am, Bm, for all 0 ≤m ≤M − 1

Output: Index set Λ containing the locations of the non zero indices of X,

estimate for sparse matrix X

1: Perform Doppler focusing for 0 ≤ i ≤ K − 1 and 0 ≤ j ≤ Q− 1:

Φ(m,ν)i,j =

P−1∑p=0

Zmi+jK,pej2πνpτ .

2: Initialization: residual R(m,ν)0 = Φ(m,ν), index set Λ0 = ∅, t = 1

3: Project residual onto measurement matrices for 0 ≤ p ≤ P − 1:

Ψν = AHRνB,

where A = [A0T A1T · · · A(M−1)T ]T , B = [B0T B1T · · · B(M−1)T ]T , and

Rν = diag(

[R(0,ν)t−1 · · · R

(M−1,ν)t−1 ]

)is block diagonal

4: Find the three indices λt = [λt(1)λt(2)λt(3)] such that

[λt(1) λt(2) λt(3)] = arg maxi,j,ν∣∣Ψν

i,j

∣∣5: Augment index set Λt = Λt

⋃{λt}

6: Find the new signal estimate

α = [α1 . . . αt]T = (ET

t Et)−1ET

t vec(Z)

7: Compute new residual

R(m,ν)t = Zm −

t∑l=1

αlej2π(− 1

2 +Λt(l,3)P

)pamΛt(l,1)

(bmΛt(l,2)

)T8: If t < L, increment t and return to step 2; otherwise stop

9: Estimated support set Λ = ΛL10: Estimated matrix XD: (ΛL(l, 2)TN + ΛL(l, 1),ΛL(l, 3))-th component is

given by αl while rest of the elements are zero

1.6.5 Cognitive SUMMeR Hardware Prototype

A cognitive SUMMeR prototype was first presented in [89]. The system realizes

a receiver with a maximum of 8 transmit (Tx) and 10 receive (Rx) antenna

elements. A scenario includes modeling of pulse transmission, accurate power

loss due to wave propagation in a realistic medium, and interaction of a trans-

mit signal with the target. A large variety of scenarios, consisting of different

target parameters, i.e., delays, Doppler frequencies, and amplitudes, and array

configurations, i.e., number of transmitters and receivers and antenna locations,

can be examined using the prototype. The waveform generator board produces

an analog signal corresponding to the synthesized radar environment, which is


Table 1.2 Technical characteristics of the cognitive SUMMeR prototype

Parameters Mode 1 Mode 2 Mode 3 Mode 4

#Tx, #Rx 8,10 8,10 4,5 8,10

Element placement Uniform Random Random Random

Equivalent aperture 8x10 8x10 8x10 20x20

Angular resolution (sine of DoA) 0.025 0.025 0.025 0.005

Range resolution 1.25 m

Signal bandwidth per Tx 12 MHz (15 MHz including guard-bands)

Pulse width 4.2 µs

Carrier frequency 10 GHz

Unambiguous range 15 km

Unambiguous DoA 180◦ (from -90◦ to 90◦)

PRI 100 µs

Pulses per CPI 10

Unambiguous Doppler from −75 m/s to 75 m/s

amplified and routed to the MIMO radar receiver board. The prototype then

samples and processes the signal in real-time. The physical array aperture and

simulated target response correspond to an X-band (fc = 10 GHz) radar.

A conventional 8x10 MIMO radar receiver would require simultaneous hard-

ware processing of 80 (or 160 I/Q) data streams. Since a separate sub-Nyquist

receiver for each of these 80 channels is expensive, we implement the eight chan-

nel analog processing chain for only one receive antenna element and serialize

the received signals of all 10 elements through this chain. This approach allows

the prototype to implement a number of receivers greater than 10 as the eight-

channel hardware only limits the number of transmitters.

If we use the same pre-filtering approach as in Section 1.3.4 for each of the

eight channels of our sub-Nyquist MIMO prototype, then the hardware design

would need a total of 4× 8 = 32 BPFs and ADCs excluding the analog filters to

separate transmit channels. We sidestep this requirement by adopting cognitive

transmission wherein the analog signal of each channel exists only in certain

pre-determined subbands and consequently, a BPF stage is not required. More

importantly, for each channel, a single low-rate ADC subsamples this narrow-

band signal as long as the subbands are coset bands so that they do not alias

after sampling [46]. This implementation needs only eight low-rate ADCs, one

per channel. Another advantage of this approach is flexibility of the prototype

in selecting the Xampling slices. Unlike Section 1.3.4, the number and spectral

locations of slices are not permanently fixed, and they can be changed.

Table 1.2 lists detailed technical characteristics of the prototype. The system

can be configured to operate in various array configurations or modes. Mode 3 and


Figure 1.17 Tx and Rxelement locations for thehardware prototypemodes over a 6 mantenna aperture. Mode4’s virtual arrayequivalent is the 20× 20ULA [18]. c©2016 IEEE.Reprinted, withpermission, from K. V.Mishra, E. Shoshan, M.Namer, M. Meltsin, D.Cohen, R. Madmoni, S.Dror, R. Ifraimov, and Y.C. Eldar, “Cognitivesub-Nyquist hardwareprototype of a collocatedMIMO radar,” inInternational Workshopon Compressed SensingTheory and itsApplications to Radar,Sonar and RemoteSensing, 2016, pp. 56-60.

Figure 1.18 Sub-NyquistMIMO prototype and userinterface. The analogpre-processor module consists oftwo cards mounted on oppositesides of a common chassis [18].c©2016 IEEE. Reprinted, with

permission, from K. V. Mishra,E. Shoshan, M. Namer, M.Meltsin, D. Cohen, R. Madmoni,S. Dror, R. Ifraimov, and Y. C.Eldar, “Cognitive sub-Nyquisthardware prototype of acollocated MIMO radar,” inInternational Workshop onCompressed Sensing Theory andits Applications to Radar, Sonarand Remote Sensing, 2016, pp.56-60.

4 are sub-Nyquist MIMO modes; the hardware switches off the inactive channels

and does not sample any data over the corresponding ADCs. Figure 1.18 shows

the sub-Nyquist MIMO prototype, user interface and radar display. As shown

in Fig. 1.19a, the cognitive radar signal occupies only certain subbands in a 15


Figure 1.19 The normalized one-sidedspectrum of one channel of a givenreceiver (a) before and (b) aftersubsampling with a 7.5 MHz ADC. Eachof the subbands spans 375 kHz and ismarked with a numeric label. In anon-cognitive processing, the signaloccupies the entire 15 MHz spectrumbefore sampling [18]. c©2016 IEEE.Reprinted, with permission, from K. V.Mishra, E. Shoshan, M. Namer, M.Meltsin, D. Cohen, R. Madmoni, S. Dror,R. Ifraimov, and Y. C. Eldar, “Cognitivesub-Nyquist hardware prototype of acollocated MIMO radar,” in InternationalWorkshop on Compressed Sensing Theoryand its Applications to Radar, Sonar andRemote Sensing, 2016, pp. 56-60.

MHz band. Here, the sliced transmit signal has eight subbands each of width 375

kHz with the frequency range of 1.63-2, 2.16-2.53, 3.05-3.42, 3.88-4.25, 5.66-6.03,

6.51-6.88, 8.64-9.01, and 12.32-12.69 MHz before subsampling. The total signal

bandwidth is 0.375 × 8 = 3 MHz. This signal is subsampled at 7.5 MHz and

the subbands locations were chosen so that there is no aliasing between different

subbands (Fig. 1.19b). A non-cognitive signal would have occupied the entire 15

MHz spectrum requiring a Nyquist sampling rate of 30 MHz. Therefore, the use

of cognitive transmission enables spectral sampling reduction by a factor of 4

(= 30 MHz/7.5 MHz) for each channel. Depending on whether the guard-bands

of the non-cognitive transmission are included in the computation or not, the

effective signal bandwidth is reduced by a factor of 5 (= 15 MHz/3 MHz) or

4 (= 12 MHz/3 MHz) respectively for each channel. Mode 3 has 50% spatial

sampling reduction when compared with Mode 1 or 2. Table 1.3 summarizes the

reduction of various resources in Mode 3 when compared with Mode 1.

We evaluated the performance of all modes through hardware experiments.

We transmitted P = 10 pulses at a PRF of 100 µs and all modes were eval-

uated against identical target scenarios. In the first experiment, when the an-

gular spacing (in terms of the sine of azimuth) between any two targets was

greater than 0.025 and the signal SNR = −8 dB, the recovery performance of

the thinned 4× 5 array in Mode 3 was not worse than Modes 1 and 2. For this

experiment, Figs. 1.20 and 1.21 show the plan position indicator (PPI) plot and

range-azimuth-Doppler maps of all the modes. Here, a successful detection (cir-

cle with light fill and no boundary) occurs when the estimated target is within

one range cell, one azimuth bin and one Doppler bin of the ground truth (circle

with dark boundary and no fill); otherwise, the estimated target is labeled as a


Table 1.3 Cognitive SUMMeR Prototype: Comparison of Resource Reduction

Resource NyquistMode 1

Sub-NyquistMode 3

Reduction

Bandwidth usage per Tx(including guard-bands)

15 MHz 3 MHz 80%

Bandwidth usage per Tx(excluding guard-bands)

12 MHz 3 MHz 75%

Temporal sampling rate perchannel

30 MHz 7.5 MHz 75%

Spatial sampling rate 8× 10 4× 5 50%Tx/Rx hardware channels 80 20 75%

Figure 1.20 Plan Position Indicator(PPI) display of results for Mode 1 and3. The origin is the location of the radar.The dark dot indicates the northdirection relative to the radar. Positive(negative) distances along the horizontalaxis correspond to the east (west) of theradar. Similarly, positive (negative)distances along the vertical axiscorrespond to the north (south) of theradar. The estimated targets are plottedover the ground truth [18, 53]. c©2017IEEE. Reprinted, with permission, fromD. Cohen, K. V. Mishra, D. Cohen, E.Ronen, Y. Grimovich, M. Namer, M.Meltsin, and Y. C. Eldar, “Sub-NyquistMIMO radar prototype with Dopplerprocessing,” in IEEE Radar Conference,2017, pp. 1179-1184.

Figure 1.21 Range-Azimuth-Dopplermap for the target configurations shownin Fig. 1.20. The lower axes representCartesian coordinates of the polarrepresentation of the PPI plots fromFig. 1.20. The vertical axis represents theDoppler spectrum [18, 53]. c©2017 IEEE.Reprinted, with permission, from D.Cohen, K. V. Mishra, D. Cohen, E.Ronen, Y. Grimovich, M. Namer, M.Meltsin, and Y. C. Eldar, “Sub-NyquistMIMO radar prototype with Dopplerprocessing,” in IEEE Radar Conference,2017, pp. 1179-1184.


Figure 1.22 Same scenario asin Fig. 5, but only Mode 3 isoperating cognitively. All modeshave the same overall transmitpower per transmitter. The insetplots show the selected region ineach PPI display on a magnifiedscale.

Figure 1.23 Same scenario asin Fig. 6, but only Mode 3 isoperating cognitively. All modeshave the same overall transmitpower per transmitter. The insetplots show the selected region ineach map on a magnified scale.

false alarm (circle with dark fill). When a target remains undetected, we label

the ground truth location as a missed detection (circle with hatched fill).

Finally, we considered a high noise scenario with SNR = −15 dB. We operated

only Mode 3 cognitively and kept all other modes in non-cognitive mode. We

noticed that the non-cognitive Nyquist 8×10 Mode 1 array exhibits false alarms

while cognitive sub-Nyquist 4 × 5 Mode 3 array is still able to detect all the

targets (Figs. 1.22 and 1.23), thereby demonstrating robustness to low SNR.

1.7 Sub-Nyquist SAR

Synthetic aperture radar (SAR) and other similar radar techniques were one

of the first applications of CS methods (see reviews in [22, 25]). SAR imaging

data are not naturally sparse in the range-time domain. However, they are often

sparse in other domains such as wavelet. Our motivation to apply sub-Nyquist

methods here is to address the following SAR processing challenge. Among the

several algorithms that are available to process SAR data, the range Doppler

1.7 Sub-Nyquist SAR 41

algorithm (RDA) is most widely used to obtain high-resolution images [90]. Its

performance is, however, limited by the range cell migration correction (RCMC)

step which requires oversampled data in order to decouple range and azimuth

axes.

Recently, [21] proposed a sub-Nyquist SAR that replaces RDA by a Fourier

domain method that achieves non-integer non-constant shifts in the RCMC in-

terpolation via the Fourier series coefficients. This avoids the interpolation step

in RCMC and further allows sub-Nyquist sampling following the Fourier-domain

analysis presented in previous sections. A similar technique was earlier employed

in ultrasound imaging [48] to dramatically reduce sampling and processing rates.

In this section, we present this Fourier domain RDA processing as a frame-

work for sub-Nyquist sampling of SAR signals. The first part of the sub-Nyquist

algorithm exploits the relationship between the signals before and after RCMC

in the Fourier domain. We show analytically, that a single Fourier coefficient

after RCMC can be computed using a small number of Fourier coefficients of

the raw data, which translates into low rate sampling as shown in Section 1.3.2.

Having the partial Fourier samples after RCMC, the second part of the algorithm

is aimed at solving a 2D CS problem in order to reconstruct the image from the

low rate samples. Finally, we then show that cognitive transmission can also be

extended to SAR. We end by demonstrating a prototype that we designed and

developed to realize concepts of cognitive SAR (CoSAR) [21].

1.7.1 Traditional SAR Processing via RDA

Consider a radar which travels along a path with velocity ν and transmits a

time-limited pulse h(t) at PRI T . The pulse has negligible energy at frequencies

beyond the bandwidth Bh/2. The transmitted pulses are sent from M different

locations, {xm}M−1m=0 , where x0 is the origin and ||xm−x0|| = m|ν|T . The pulses

are transmitted into a scene with reflectivity σ(r). The received signal, after

coherent demodulation, is given by

dm(t) =

∫σ(r)h(t− 2||r− xm||/c)× wa(xm, r)e−j4πfc||r−xm||/cdr, (1.59)

where ||r−xm|| is the distance from the radar to a scatter point and wa(xm, r) is

the antenna beam pattern which varies depending on the SAR operation mode

[90]. The main goal of SAR data processing is to construct the scene’s reflectivity

map, σ(r), from the raw data. The reflected signal dm(t) at a point m requires to

be sampled at least at the bandwidth Bh as per the Nyquist sampling theorem.

The resulting discrete-time signal is d[n,m] = dm(nTs), with 0 ≤ n < N =

bTfsc, where fs = 1/Ts is the sampling rate.

RDA processing consists of the following steps. First the sampled raw data is

compressed in the range dimension:

s[n,m] = d[n,m] ∗ h∗[−n], (1.60)

where h[n] is the sampled transmit signal. This data is then transformed to the


range-Doppler domain using DFT along the azimuth:

S[n, k] =

M−1∑m=0

s[n,m]e−j2πkm/M . (1.61)

RCMC is applied assuming a far-field approximation. The purpose of RCMC is

to compensate for the effect of range cell migration due to the varied satellite-

scatterer distance and to correct the hyperbolic behavior of the target trajecto-

ries. The RCMC operator can be written as

C[n, k] = S[n+ nak2, k]. (1.62)

For every Doppler frequency k, the range axis is scaled by 1+ak2. The value of a

is predetermined depending on the observation mode. For example, in stripmap

SAR, a = λ2

8|ν|T 2M2 . This range-variant shift requires values which fall outside

the discrete grid. A MF then achieves compression in azimuth via

Y [n, k] = C[n, k]e−jπk2

Ka[n] , (1.63)

where Ka[n] is the range dependent azimuth chirp rate. Finally, an inverse DFT

in the azimuth direction yields the focused image:

I[n,m] =1

M

M−1∑k=0

Y [n, k]ej2πmk/M . (1.64)

There are two ways to implement RCMC: In the first option, RCMC is per-

formed by range interpolation in the Range-Doppler domain. However, this in-

terpolation is time-consuming and computationally demanding. The second ap-

proach involves the assumption that the range cell migration is range invariant,

at least over a finite range block. In this case, RCMC is implemented using a

DFT, linear phase multiply, and inverse DFT per block. However, this imple-

mentation has the disadvantage that blocks should overlap in range, and the

efficiency gain may not be worth the added complexity.

1.7.2 Fourier domain RDA and sub-Nyquist SAR

In this section, we introduce a new RDA processing technique implemented in

frequency using the Fourier series coefficients of the raw data. This paves the way

for substantial reduction in the number of samples in the time-domain interpo-

lation needed to obtain the same image quality and without any assumptions on

the signal structure or the invariance of range blocks.

We begin with the continuous version of (1.62):

Ck(t) = Sk(t(1 + ak2)), (1.65)

where Sk(nTs) = S[n, k]. The Fourier series coefficients of Ck(t) over the interval

[0, T ) can be expressed as [21]

Ck[l] =1

T

T∫0

Sk(t)qk,l(t), (1.66)

where qk,l(t) approximates the scaling operation in (1.62). The Fourier coeffi-

1.7 Sub-Nyquist SAR 43

cients of the continuous-time signals Sk(t) and qk,l(t) are, respectively, Sk[n]

and

Qk,l[n] =1

1 + ak2e−jπ(n+ 1

1+ak2 )sinc

(n+

l

1 + ak2

). (1.67)

It can be shown that most of the energy of the set Qk,l[n] is concentrated around

a specific component nk,l. Thus, for every Doppler frequency k, the Fourier series

coefficients of the scaled signal, Ck(t), can be calculated as a linear combination

of a local choice of Fourier series coefficients of Sk(t)

Ck[l] =∑

n∈ν(k,l)

Sk[n]Qk,l[−n], (1.68)

where ν(k, l) is the set of indices that dictate the decay property of Qk,l[n].

Assuming the Fourier series coefficients Dm[l] of the raw data dm(t) can be

acquired directly, the range compression is achieved in the Fourier domain as

Dm[l] = TDm[l]H∗[l], (1.69)

where H[l] are the Fourier series coefficients of the transmitted pulse h(t). Ap-

plying azimuth DFT gives Sk[l] that can be used in (1.68) to perform Fourier

domain RCMC. The inverse DFT on the coefficients Ck[l] provides the corrected

sampled signal after RCMC. One could then proceed with the remaining steps

of RDA, i.e., (1.63) and (1.64), to complete the processing.

The number of Fourier coefficients required can be further reduced if a basis

(e.g. wavelet) is found in which the desired image is sparse. Then, the relation-

ship between Ck[l] and the raw data samples Dm[l] can be exploited to solve for

the coefficients in the sparse basis using fewer Fourier coefficients. In [21], it was

suggested to modify fast iterative shrinkage-thresholding algorithm (FISTA) to

solve this problem and achieve full data reconstruction from the partial measure-

ments with reasonable computational load. Similar to cognitive pulse Doppler

radar (Section 1.5.2) and cognitive SUMMeR (Section 1.6.5), sub-Nyquist SAR

systems can also be modified to fit cognitive radar requirements and allow for dy-

namic transmission and reception of several narrow frequency bands. We present

the hardware prototype of such a system in the next subsection.

1.7.3 Hardware Prototype

We designed and developed a hardware prototype of a CoSAR system and eval-

uated Fourier domain RDA processing in real-time. Figure 1.24 shows the entire

set up. The PRI is 51.2 µs and carrier frequency of the signal is 90 MHz. A con-

trol interface (Fig. 1.25) activates the prototype which generates the desired I/Q

signal and feeds it to the analog pre-processor (inset). The analog pre-processor

filters have 30 dB stopband attenuation in order to filter out interference from

neighboring channels. The digital receiver obtains and processes samples at low-

rates. The processed image is then shown on the radar display. We used a 5 MHz

cognitive chirp signal whose only 4 narrow subbands of 625 kHz bandwidth were


Figure 1.24 Cognitive SAR (CoSAR)prototype and (inset) analogpre-processor.

Figure 1.25 CoSAR GUI showing thecognitive and non-cognitive chirpwaveforms along with the sampledsubbands at top right.

sampled and processed by the digital receiver. The Xampling and RCMC are

performed at 1/4th and 1/8th of the Nyquist rate, respectively.

Similar to the cognitive SUMMeR system, our CoSAR prototype can operate

in both cognitive and non-cognitive modes. Figure 1.26 shows results of these

modes at Nyquist and sub-Nyquist sampling rates at SNR = 2dB. The range and

cross-range (azimuth) resolutions are 30 and 10 m, respectively. When compared

with the Nyquist rate of 10 MHz, the combined sampling rate of the four slices is

2.5 MHz leading to reduction of rate by 75%. We note that CoSAR reconstruction

exhibits smaller error than the non-cognitive Nyquist processing in low SNR

scenarios despite sampling at a significantly reduced sampling rate. Further, the

prototype demonstrates operation of SAR using narrow subbands that can be

adaptively changed. This opens up the possibility of spectral coexistence of SAR

with other satellite-borne services.

Figure 1.26 Comparison ofprototype outputs for an imageof a ship.

1.8 Summary 45

1.8 Summary

In this chapter, we reviewed sub-Nyquist radar principles, algorithms, proto-

types, and applications. Our focus was on pulse-Doppler systems for which sub-

Nyquist processing can be individually applied to temporal, Doppler, and spatial

domains. Our approach has distinct advantages over several past CS-based de-

signs. The proposed sub-Nyquist radar receivers perform low rate sampling and

processing which can be implemented with simple hardware, impose no restric-

tions on the transmitter, use a CS dictionary that does not scale up with the

problem size, and exhibit robustness to clutter and noise.

We presented colocated MIMO radar as an application where joint spatio-

temporal sub-Nyquist processing leads to reduction in antenna elements and sav-

ings in signal bandwidth. In SAR imaging, sub-Nyquist processing in the Fourier

domain leads to sampling rate reduction without compromising high-quality and

high-resolution imaging. We demonstrated that sub-Nyquist receivers lead to the

feasibility of cognitive radar which transmits thinned spectrum signals. This de-

velopment was significant in making the spectral coexistence of radar with a

communication service possible. We also extended cognition ideas based on sub-

Nyquist processing to MIMO and SAR systems.

Most importantly, we emphasized that sub-Nyquist radars are realizable in

hardware for each of the systems described in this chapter. The hardware pro-

totypes were in-house and custom-made using many off-the-shelf components.

The systems operate in real-time and their performance is robust to high noise

and clutter. We believe that such practical implementations pave the way to

delivering the promise of reduced-rate processing in radar remote sensing.

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